High School Algebra I Unlocked (2016)
Chapter 2. Exponents and Sequences
Lesson 2.2. Sequences
Think about the last time you listened to a song on the radio, computer, phone, or some other music-providing device. A song is a specific sequence of notes that work together to form something that sounds good. If the notes were randomly put together in a different order, the song would probably not be very popular. After all, no one enjoys listening to a random mish-mash of discordant notes.
In math, a sequence is a series of numbers that are arranged in a specific order according to a rule, which is the pattern of a sequence that allows us to determine unknown terms. There are two types of sequences: finite and infinite. Afinite sequence is a sequence that has a limited number of terms, while an infinite sequence has an unlimited number of terms and is indicated by three dots after the last number (…) in a set.
This question requires you to find the value of x that satisfies the equation 27^{x} ^{+ 2} = . In order to properly use the rules of exponents, you first need to put all terms in the same base—in this case, base 3. You will find that 27^{x} ^{+ 2} = becomes (3^{3})^{x} ^{+ 2} = , and finally 3^{3x} ^{+ 6} = . Next, use the division rules of exponents to eliminate the fraction on the right side of the equation; 3^{3x} ^{+ 6} = becomes 3^{3x} ^{+ 6} = 3^{2 − 2x} ^{+ 6}. Now that all the bases are the same, you can set the exponents equal to one another: 3x + 6 = 2 − 2x + 6. Finally, simplify the equation and solve for x. 3x + 6 = 2 − 2x + 6, 5x + 6 = 8, 5x = 2, and x = 2/5. Therefore, the real value of x that satisfies the equation 27^{x} ^{+ 2} = is 2/5, or (B).
Finite Sequences |
Infinite Sequences |
Sequence Rule |
{1, 2, 3, 4} |
{1, 2, 3, 4, …} |
Add 1 to the previous term. |
{2, 4, 6, 8, 10, 12} |
{2, 4, 6, 8, 10, 12, …} |
Add 2 to the previous term. |
{100, 90, 80} |
{100, 90, 80, …} |
Subtract 10 from the previous term. |
{8, 4, 2, 1, } |
{8, 4, 2, 1, , …} |
Divide the previous term by 2. |
{2, 4, 8, 16, 32} |
{2, 4, 8, 16, 32, …} |
Multiply the previous term by 2. |
{0, 1, 0, 1, 0, 1, 0} |
{0, 1, 0, 1, 0, 1, 0, …} |
Alternating 0s and 1s. |
The chart shows a number of different sequences and the rules associated with each one. You may have noticed that sequences use the same notation as sets: elements in a sequence are separated by commas and placed within a set of curly braces. Unlike in a set, however, the order of elements in a sequence is extremely important; the order determines the rule of the sequence. Furthermore, elements in a sequence can appear multiple times, such as in the final example in the chart.
Let’s try a couple of sequence questions.
EXAMPLE
What is the next term in the sequence below?
{3, 9, 27, …}
Here you’re asked to find the next term in the sequence {3, 9, 27, …}. You need to identify the rule of the sequence, or the pattern behind the numbers. Take a look at the first two numbers, 3 and 9. These terms are related in lots of ways, so let’s write down the different relationships:
Terms |
Relationship |
Relationship |
Relationship |
3 → 9 |
Multiply term by 3 |
Square term |
Add 6 to term |
Now find the relationships for the second and third terms, 9 and 27.
Terms |
Relationship |
Relationship |
9 → 27 |
Multiply term by 3 |
Add 18 to term |
Next, take a look at the relationships at the same time and eliminate any relationships that don’t apply to both sets of terms.
Terms |
Relationship |
Relationship |
Relationship |
3 → 9 |
Multiply term by 3 |
Square term |
Add 6 to term |
9 → 27 |
Multiply term by 3 |
Add 18 to term |
The only relationship that applies to both sets of numbers is to multiply by 3, so to find the next number in this sequence, you need to multiply the previous term by 3. Therefore, the next number in the sequence {3, 9, 27, …} is 3 × 27 = 81.
Often, you will find that there is more than one rule that applies to the terms in a sequence. If you are asked to provide the rule, stick with the most straightforward one, while noting that other rules do exist.
Here is how you may see fractional exponents on the SAT.
Which of the following accurately rewrites the expression in the form A · B^{t}?
A)
B)
C)
D)
EXAMPLE
What is the sixth term in the sequence below?
{10, −20, 40, −80, …}
As in Example 1, you need to identify the rule, or the pattern, of the sequence, so determine the different relationships:
Terms |
Relationship |
Relationship |
Relationship |
10 → −20 |
Subtract 30 |
Multiply by −2 |
Subtract 3 times the term |
−20 → 40 |
Add 60 |
Multiply by −2 |
Add −3 times the term |
40 → −80 |
Subtract 120 |
Multiply by −2 |
Subtract 3 times the term |
Now look at the relationships concurrently, eliminating any relationships that don’t apply to all sets of terms. Here, the terms are related by a factor of −2.
Terms |
Relationship |
Relationship |
Relationship |
10 → −20 |
Subtract 30 |
Multiply by −2 |
Subtract 3 times the term |
−20 → 40 |
Add 60 |
Multiply by −2 |
Add −3 times the term |
40 → −80 |
Subtract 120 |
Multiply by −2 |
Subtract 3 times the term |
Thus, in order to find the fifth term of this sequence, multiply −80 by −2 to find that −80 × −2 = 160. Repeat the process with the fifth term to find that the sixth term of the sequence is 160 × −2 = −320.
Before we leave Example 11 entirely, take a look back at the relationships that we eliminated. You may have noticed that all of the relationships here have a pattern. For example, the first column shows that the terms are related by alternatively subtracting and adding twice the difference between the two terms; to move from 10 → −20 subtract 30, to move from −20 → 40 add 60, to move from 40 → −80 subtract 120, to move from −80 → 160 add 240, and to move from 160 → −320 subtract 480.
The last column also shows a distinct relationship: alternate between subtracting 3 times the term and adding −3 times the term; 10 → −20 is represented as 10 − (3 × 10) = −20, −20 → 40 is represented as −20 + (−3 × −20) = 40, and 40 → −80 is represented as 40 − (3 × 40) = −80, −80 → 160 is represented as −80 + (−3 × −80) = 160, and 160 → −320 is represented as 160 − (3 × 160) = −320.
It is entirely possible for a sequence to have more than one rule. The key, however, is to ensure that the rule applies to all known elements in a sequence. If you find that there is more than one rule that applies to a sequence, work with the most basic rule.
Arithmetic Sequences
Sequences have a specific notation to make them more mathematically accessible. A specific term in a sequence is written as
x_{n}
The notation indicates that x_{n} is the value of the term and n is the index, or the number of the term in the sequence. You can use this sequence to write the rules of a sequence algebraically. The following chart is from earlier in this chapter, but it has one change: The rule associated with each sequence is now written in sequence notation.
To put the expression in the form A · B^{t} you need to separate the t exponents from the rest of the expression. To do so, remember your MADSPM exponent rules. When you divide, you subtract exponents, so the expression 3^{2t} ^{− 1} in the numerator is the equivalent of . You multiply exponents when you have parentheses, and you want to isolate the t as the only exponent in the expression to get to A · B^{t} form. Therefore, you can rewrite 3^{2t} as (3^{2})^{t}, or 9^{t}. So, 3^{2t} ^{− 1} = . Similarly, in the denominator, 2^{t} ^{+ 1} = 2^{t} + 2.
With these factors in mind, we can turn to the original expression. Use the above equations and substitute in to the expression:
Move the 3 in the denominator of to the denominator of the primary fraction and rearrange the terms so the terms to the power of t are to the right of the terms that are to no power:
Finally, multiply terms to no power together and rewrite everything as the product of two fractions:
This is (D).
Finite Sequences |
Infinite Sequences |
Sequence Rule |
{1, 2, 3, 4} |
{1, 2, 3, 4, …} |
x_{n} = n + 1 |
{2, 4, 6, 8, 10, 12} |
{2, 4, 6, 8, 10, 12, …} |
x_{n} = n + 2 |
{100, 90, 80} |
{100, 90, 80, …} |
x_{n} = n − 10 |
{8, 4, 2, 1, } |
{8, 4, 2, 1, , …} |
x_{n} = n × |
{2, 4, 8, 16, 32} |
{2, 4, 8, 16, 32, …} |
x_{n} = n × 2 |
{0, 1, 0, 1, 0, 1, 0} |
{0, 1, 0, 1, 0, 1, 0, …} |
There are two major classifications of sequences that you need to be aware of: arithmetic sequences and geometric sequences. In an arithmetic sequence, the difference between one term and the next is a constant, signified by the variable d.
The arithmetic sequence of the set {1, 2, 3, 4, 5, …} is written algebraically as follows:
{a, a + d, a + 2d, a + 3d, a + 4d, …}
In this sequence, a is the first term in the sequence, and d is the common difference, or the difference between the terms.
You may encounter
questions that refer to
arithmetic or geometric
progressions. Progressions
are another name for
sequences. You’ll solve
progression questions
in the same way as you
would sequence questions.
EXAMPLE
If S = {3, 5, 7, 9, 11}, what is the value of S_{4} ?
This question requires you to understand sequence notation, in which n in the term S_{n} represents the index, or term number, in a sequence, S. Since the question asks you to find the value of S_{4}, you need to find the fourth term in the sequence. In the sequence {3, 5, 7, 9, 11}, S_{4} = 9.
While some questions may simply require you to understand sequence notation, others will require you to determine the rule of the sequence and then find future terms.
EXAMPLE
If S = {3, 5, 7, 9, 11, …}, what is the value of S_{9} ?
Unlike the previous example, which was based solely on your ability to understand sequence notation, this question requires you to find the rule, expand the sequence, and determine the eighth term in the sequence. First, identify the relationship between the elements. In order to move between one term and the next, two must be added to the previous term; for example, 3 + 2 = 5, 5 + 2 = 7, and 7 + 2 = 9. Once you’ve determined the rule, you can find the next terms.
Start by finding the value of S_{6}:
S_{6} = S_{5} + 2
= 11 + 2
= 13
Now find the value of S_{7}:
S_{1} = S_{6} + 2
= 13 + 2
= 15
Next, find the value of S_{8}:
S_{8} = S_{7} + 2
= 15 + 2
= 17
Finally, you can find the value of S_{9}:
S_{9} = S_{7} + 2
= 17 + 2
= 19
Therefore, if S = {3, 5, 7, 9, 11, …}, the value of S_{9} = 19.
However, you may not always be able to write all of the terms out. Since arithmetic sequences have common differences, you can use a shortcut. In this example, our common difference is 2, we know the value of S_{5}, and we want to find the value of S_{9}. There are four terms between S_{5} and S_{9}, so there is a total difference of 2 × 4 = 8 between the two elements.
Now you can find S_{9}:
S_{9} = S_{5} + 8
= 11 + 8
= 19
Just like before, if S = {3, 5, 7, 9, 11, …}, the value of S_{9} = 19. However, this method of tackling arithmetic sequences is much quicker than listing out all the elements one by one.
Geometric Sequences
Unlike in an arithmetic sequence, each term in a geometric sequence is found by multiplying the previous term by a constant, known as the common ratio, signified by the variable r. The geometric sequence of the set {2, 4, 8, 16, 32, …} is written as follows:
{a, ar, ar^{2}, ar^{3}, ar^{4}, …}
In this sequence, a is the first term in the sequence, and r is the common ratio, or the factor between the terms. To find the value of any element in a geometric sequence, remember the geometric rule x_{n} = ar^{(n} ^{− 1)} = ar^{k}.
Therefore, we could classify the finite and infinite sequences as follows:
Finite Sequences |
Infinite Sequences |
Sequence Rule |
Sequence Type |
{1, 2, 3, 4} |
{1, 2, 3, 4, …} |
x_{n} = n + 1 |
arithmetic |
{2, 4, 6, 8, 10, 12} |
{2, 4, 6, 8, 10, 12, …} |
x_{n} = n + 2 |
arithmetic |
{100, 90, 80} |
{100, 90, 80, …} |
x_{n} = n − 10 |
arithmetic |
{8, 4, 2, 1, } |
{8, 4, 2, 1, , …} |
x_{n} = ar^{(n} ^{− 1)} |
geometric |
{2, 4, 8, 16, 32} |
{2, 4, 8, 16, 32, …} |
x_{n} = ar^{(n} ^{− 1)} |
geometric |
EXAMPLE
If S = {3, 9, 27, 81, …}, what is the value of S_{10} ?
First determine whether this is an arithmetic or geometric sequence. Is there a common difference between the terms or a common ratio? If you said there is a common ratio, you’re right!
Remember that a geometric sequence is represented in the form of {a, ar, ar^{2}, ar^{3}, …}, where a is the first term and r is the common ratio. In this case, the first term a is 3, and the common ratio between the terms, r, is also 3.
Now that you have the value of both a and r, plug into the geometric rule x_{n} = ar^{(n} ^{− 1)} and simplify the equation to find that
S_{10} = ar^{(n} ^{− 1)}
= (3)(3)^{(10 − 1)}
= (3)(3)^{9}
= 3^{10}
= 59,049
Therefore, the value of S_{10} in the geometric sequence {3, 9, 27, 81, …} is 59,049.
When working with
geometric sequences the
value of r will not be zero.
Think about it. If r = 0,
the resulting sequence
will be {a, 0, 0, 0, 0, …},
which is not geometric!
Sequence Summations
You may also encounter questions that require you to find the sum of elements in a sequence. To do this, you will use sigma notation, which is mathematical shorthand used to find sums. The sigma notation that represents the sum of the first five elements of the arithmetic sequence {1, 2, 3, 4, 5, …} is
This notation indicates that you are summing the value of n from n = 1 to n = 5:
= 1 + 2 + 3 + 4 + 5 = 15
However, you may not always have the elements of a sequence readily available. In order to find the sum of any number of terms in any arithmetic sequence, use the following formula:
In this sigma notation, k starts at 0 (as the first element in a sequence occurs when k = 0) and goes up to n − 1, a is the first term, d is the common difference, and n is the number of terms to sum. Looking at the sequence {1, 2, 3, 4, 5, …}, you can plug in a = 1 and d = 1 for n = 5 elements to find the sum of the first five elements:
Again, we find that the sum of the first five elements of the sequence {1, 2, 3, 4, 5, …} is 15. Remember, using the arithmetic sequence summation formula can transform tricky questions into more straightforward ones!
Sigma notation can also be used to find the sums of portions of a sequence. For example, if you were asked to find the sum of the third and fourth elements of a sequence having the rule n^{2} + 1, the sigma notation would be the following:
(n^{2} + 1) = (9 + 1) + (64 + 1) = 75
Now let’s try a couple questions dealing with summations in arithmetic sequences.
EXAMPLE
Find the value of ?
Sigma notation indicates that you will be finding the sum of terms in a sequence. The notation here indicates that you can find the value of each term in the sequence by plugging in n to the formula . In order to find the sum of the first ten terms of this sequence, you will use the following formula for arithmetic sequences:
In the formula, k starts at 0 and goes up to n − 1, a is the first term, d is the common difference, and n is the number of terms to sum. Before we can use the formula, however, we need to determine a and d.
Start with the first term, to find S_{1}:
Now find the second term, S_{2}, by setting n = 2:
Because the difference between terms in an arithmetic sequence is constant, you can now find the value of d:
d = s_{2} − s^{1}
d = − 1
d =
Now plug the values a = 1 and d = into the following formula to find the sum of the first ten terms of the sequence:
The sum of the first 10 terms of the arithmetic sequence with the rule is 32.5.
You can use a slightly different approach to find a specific element or the sum of multiple elements in a geometric sequence. For example, to find a single element in a geometric sequence, you would use the following formula:
ar^{n} ^{− 1} = ar^{k}
This notation indicates that a term in the sequence is ar^{n} ^{− 1} = ar^{k}, where k is the term you are solving for, a is the first term, and r is the common ratio. Consider the sequence {2, 4, 8, 16, 32, …}. You can plug in a = 2 and r = 2 repeatedly to find the first five elements in the sequence separately. Then, find the sum of the elements.
Thus, the sum of the first five elements of the geometric sequence {2, 4, 8, 16, 32, …} is 62. However, this may not always be the most effective approach. What if you don’t know all the numbers in the geometric sequence? Or what if you are asked to find the sum of the first 100 elements in a sequence? You likely won’t have the time or patience to sit and write out each term and sum them all up. Instead, use the following formula to find the sum of any number of elements in a geometric set, where k starts at 0 and goes up to n − 1, a is the first term, r is the common ratio, and n is the number of terms to sum.
Here is how you may see arithmetic sequences on the ACT.
What is the sum of the first three terms of the arithmetic sequence in which the 7th term is 13.5, and the 11th term is 18.3 ?
F. 15.9
G. 22.5
H. 25.5
J. 32.4
K. 43.5
Let’s see how this formula works to find the sum of the first five elements of the geometric sequence {2, 4, 8, 16, 32, …}.
EXAMPLE
What is the sum of the 11th and 12th terms of a geometric sequence in which the first term is 4 and the common ratio is ?
In order to find the sum of the 11th and 12th terms of a geometric sequence, you can sum the values of the 10th and 11th terms in the sequence.
Since the question states that the first term is 4 and the common ratio is , you can plug a = 4 and r = into the equation for finding individual terms as follows:
Thus, the sum of the 11th and 12th terms of a geometric sequence in which the first term is 4 and the common ratio is 1/2 is approximately 0.0059.
Again, as long as you use the correct formula, you can transform tricky sequence questions into more manageable ones.
To solve this problem, you can use the formula for arithmetic sequences, (a + kd) = (2a + (n − 1) d), where k starts at 0 and goes up to n − 1, a is the first term, d is the common difference, and n is the number of terms to sum. In order to find the common difference, subtract the value of the 7th term from the 11th term and divide the value by 4 (to account for the four terms between the 7th and 11th term): 18.3 − 13.5 = 4.8, 4.8/4 = 1.2. Now you need to solve for a. If the common difference is 1.2 and the 7th term in the sequence is 13.5, the first term in the sequence will be 13.5 − 6(1.2) = 13.5 − 7.2 = 6.3. Finally, plug in a = 6.3 and d = 1.2 for n = 3 terms into the formula (a + kd) = (2a + (n − 1) d) to find the sum of the first three terms. You will find that (a + kd) = (2(6.3) + (3 − 1)1.2), (12.6 + (2)1.2), (12.6 + 2.4), and 18.9 + 3.6 = 22.5, which is (G).
Nice job! You are now an exponent expert and a skilled sequencer and, really, what more could you ask for? Review questions to practice your new skills, perhaps? Your wish is granted. Try the questions on the next page to see if you can master them all!
DRILL
CHAPTER 2 PRACTICE QUESTIONS
Click here to download a PDF of Chapter 2 Practice Questions.
Directions: Complete the following problems as specified by each question. For extra practice, try using an alternative method to solve the problem or check your work.
1. Which of the following is equivalent to 55,125 ?
A) 3^{3} × 5^{2} × 7^{2}
B) 3^{2} × 5^{2} × 7^{3}
C) 3^{2} × 5^{3} × 7^{3}
D) 3^{2} × 5^{3} × 7^{2}
2. Simplify the expression (2a^{2} + b^{2})^{5}(2a^{2} + b^{2})^{−5}.
3. What are the first three terms of the sequence a_{n} = 2^{n} ^{− 1} ?
4. Evaluate the expression (4x^{2}y^{4})^{}.
5. Simplify the expression .
6. What is the sum of the first three terms of the geometric sequence a_{n} = (2^{n})n ?
7. Simplify the following expression:
8. What is the value of the arithmetic sequence (11n − 6) ?
9. If 64^{x} ^{+ 4} = 16^{x} ^{+ 7}, what is the value of x ?
10. What is the sum of the 16th and 19th terms of a geometric sequence in which the first term is and the common ratio is 2 ?
SOLUTIONS TO CHAPTER 2 PRACTICE QUESTIONS
1. D
In order to identify the expression that is equivalent to 55,125, create a factor tree to determine the prime factorization of the number. (See figure.) Once you’ve created your factor tree, you will find that 55,125 = 3 × 3 × 5 × 5 × 5 × 7 × 7 = 3^{2} × 5^{3} × 7^{2}. Thus, the correct answer is (D).
2. 1
To answer this question, apply the rules of exponents to the expression while following the order of operations. Although the expression looks complicated, notice that the information inside the parentheses is identical in each term. Therefore, the two terms have the same base, different exponents, and are multiplied together. When two terms with identical bases are multiplied together, the exponents are added. Thus, (2a^{2} + b^{2})^{5}(2a^{2} + b^{2})^{−5} = (2a^{2} + b^{2})^{5 + (−5)} = (2a^{2} + b^{2})^{0}. Any term to the power of zero is equal to 1, so (2a^{2} + b^{2})^{0} = 1.
3. 1, 2, and 4
In order to find the first three terms of the sequence a_{n} = 2^{n} ^{− 1}, you need to find a_{1}, a_{2}, and a_{3} by changing the value of n. Starting with the first term, to find S_{1}:
S_{1} = 2^{n} ^{− 1}
= 2^{1 −1}
= 2^{0}
= 1
Now find the second term, S_{2}, by setting n = 2:
S_{2} = 2^{n} ^{− 1}
= 2^{2 −1}
= 2^{1}
= 2
Finally, find the third term, S_{3}, by setting n = 3:
S_{3} = 2^{n} ^{− 1}
= 2^{3 −1}
= 2^{2}
= 4
Therefore, the first three terms of the arithmetic sequence a_{n} = 2^{n} ^{− 1} are 1, 2, and 4.
4. 2xy^{2}
Here you’re asked to evaluate the expression (4x^{2}y^{4})^{}. In order to answer this question, you need to know how to work with fractional exponents; namely, that a term in the form . Accordingly, the expression (4x^{2}y^{4})^{} = . Now simplify the expression as you would any other expression under a radical sign. Thus, = 2xy^{2}.
5. 8ab^{33}
This question requires you to simplify the exponential expression according to the rules of exponents. Start by simplifying the numerator of the expression, recalling that you multiply exponents together when a power is raised to a power.
Next, you can simplify the expression further, eliminating the a-term in the denominator by dividing exponents.
= 8ab^{33}
And you’re done! The expression can be simplified as 8ab^{33}.
6. 34
Here you are asked to find sum of the first three terms of the sequence a_{n} = (2^{n})n, indicating that you need to find a_{1}, a_{2}, and a_{3} by changing the value of n. Start by finding the first term, S_{1}:
S_{1} = (2_{n})n
= (2^{1})1
= 2^{1}
= 2
Now find the second term, S_{2,} by setting n = 2:
S_{2} = (2_{n})n
= (2^{2})2
= (4)2
= 8
Finally, find the third term, S_{3,} by setting n = 3:
S_{3} = (2_{n})n
= (2^{3})3
= (8)3
= 24
Therefore, the first three terms of the arithmetic sequence a_{n} = (2^{n})n are 2, 8, and 24, and the sum of the first three terms is 2 + 8 + 24 = 34.
7. or 8d^{9}c^{−7}
Here, you’re asked to simplify the expression . Although there are many things happening in this question, if you apply the rules of exponents, you’ll be fine! Start by simplifying the first term in the expression, recalling that you multiply exponents together when a power is raised to a power.
Next, put all of the numerical terms into the same base, and then simplify each term by dividing exponents with similar bases. Remember that when dividing exponential terms, the exponents are subtracted from one another.
Then multiply the two terms together, remembering to add the exponents:
Finally, divide the remaining terms to find the simplified expression.
or 8d^{9} c^{−7}
8. 107,730
In this question you’re asked to find the value of (11n − 6), or the sum of the first 140 terms of this sequence. Therefore, you will use the following formula for arithmetic sequences:
(a + kd) = (2a) + (n − 1) d)
In the formula, k starts at 0 and goes up to n − 1, a is the first term, d is the common difference, and n is the number of terms to sum. However, before you can use the formula, you need to determine a and d.
Starting with the first term, find S_{1}:
S_{1} = 11n − 6
= 11(1) − 6
= 5
Now find the second term, S_{2,} by setting n = 2:
S_{2} = 11n − 6
= 11(2) − 6
= 22 − 6
= 16
Since the difference between terms in an arithmetic sequence is constant, you can now find the value of d:
d = S_{2} − S_{1}
d = 16 − 5
d = 11
Next, plug the values a = 5, n = 140, and d = 11 into the following formula to find the sum of the first 140 terms of the sequence:
(a + kd) = (2a + (n − 1)d)
= (2(5) + (140 − 1)11)
= 70(10) + (139)11)
= 70(10 + 1,529)
= 70(1,539)
= 107,730
Therefore, the sum of the first 140 terms of the arithmetic sequence with the rule 11n − 6 is 107,730.
9. 2
This question requires you to find the value of x, given that 64^{x} ^{+ 4} = 16^{x} ^{+ 7}. In order to solve for x, you must first rewrite each term in a common base. In this scenario, the common base is 4.
64^{x} ^{+ 4} = 16^{x} ^{+ 7}
4^{3})^{x} ^{+ 4} = (4^{2})^{x} ^{+ 7}
Next, simplify the expression, remembering that exponents are multiplied together when an exponent is raised to a power.
(4^{3})^{x} ^{+ 4} = (4^{2})^{x} ^{+ 7}
4^{3x} ^{+ 12} = 4^{2x} ^{+ 14}
Now that the bases are identical, set the exponents equal to one another and solve for x:
4^{3x} ^{+ 12} = 4^{2x} ^{+ 14}
3x + 12 = 2x + 14
3x − 2x = 14 − 12
x = 2
Therefore, given that 64^{x} ^{+ 4} = 16^{x} ^{+ 7}, x = 2.
10. 147,456
Here you are told that the first term in a geometric sequence is 1/2, the common ratio is 2, and you need to find the sum of the 16th and 19th terms of the sequence. Therefore, use the general formula to find the value of a term in a geometric sequence, ar^{k}, to identify the 16th and 19th terms in this sequence, setting a = 1/2 and r = 2.
ar^{15} + ar^{18} = (2)^{15} + (2)^{18}
=
= 2^{14} + 2^{17}
= 16,384 + 131,072
= 147,456
Therefore, the sum of the 16th and 19th terms of a geometric sequence in which the first term is 1/2 and the common ratio is 2 is 147,456.
REFLECT
Congratulations on completing Chapter 2!
Here’s what we just covered.
Rate your confidence in your ability to:
•Simplify exponents using the rules of exponents
1 2 3 4 5
•Manipulate and work with negative exponents
1 2 3 4 5
•Explain the special rules of exponents
1 2 3 4 5
•Complete the prime factorization of large numbers
1 2 3 4 5
•Solve expressions with non-integer exponents
1 2 3 4 5
•Identify equivalent exponential expressions
1 2 3 4 5
•Identify patterns of sequences, series, and summations
1 2 3 4 5
•Solve both arithmetic and geometric sequences
1 2 3 4 5
If you rated any of these topics lower than you’d like, consider reviewing the corresponding lesson before moving on, especially if you found yourself unable to correctly answer one of the related end-of-chapter questions.
CHAPTER 2 KEY POINTS
The exponent associated with a base number tells you how many times you should multiply the base number by itself.
The value of a negative exponent is equal to the reciprocal of the positive exponent.
Convert large numbers to exponents by identifying the prime factors of a number.
When multiplying exponents, add them together: x^{3} × x^{4} = x^{7}.
When dividing exponents, subtract them from one another: = x^{3}.
When you have an exponent raised to a power, multiply the exponents together; (x^{3})^{5} = x^{15}.
A radicand is the number that is under the radical symbol and the index tells you the degree of the root; in , k is the radicand and n is the index.
A sequence is a series of numbers that are arranged in a specific order according to a rule or pattern.
A finite sequence is a sequence that has a limited number of terms, while an infinite sequence has an unlimited number of terms.
Sequence notation states that x_{n} is the value of the term and n is the index, or the number of the term in the sequence.
In an arithmetic sequence, the difference between one term and the next is a constant.
To find a single term in an arithmetic sequence, use the formula a_{n} = a + (n − 1)d.
In a geometric sequence, each term is found by multiplying the previous term by a common ratio.
To find a single term in a geometric sequence, use the formula ar^{n} ^{− 1} = ar^{k}.
Use sigma notation to find the sum of the terms in a sequence.
In order to find the sum of any number of terms in any arithmetic sequence, use the formula (a + kd) = (2a + (n − 1)d).
In order to find the sum of any number of terms in any geometric sequence, use the following formula .