9. Sequences, Induction, and Probability

Find the Value of an Annuity

Geometric Sequences and Series

Determine the Future Value of a Savings Annuity and Interest Earned (Monthly)

Exercises 88–90 will help you prepare for the material covered in the next section. Consider the sequence whose nth term is an = (3)5^n Find a2/a3, a1/a2, a4/a3 and a5/a4 What do you observe?

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30.

Find the sum of the first 12 terms of the geometric sequence: 2, 6, 18, 54, ...

In Exercises 45–50, express each repeating decimal as a fraction in lowest terms.

0.5 ̅ (repeating 5) = 5/10 + 5/100 + 5/1000 + 5/10,000 + ...

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a(sub 7) when a(sub 1) = 2, r = 3

In Exercises 17–24, write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a7, the seventh term of the sequence.

3, 12, 48, 192, ...

In Exercises 45–50, express each repeating decimal as a fraction in lowest terms.

0.47 ̅ (repeating 47) = 47/100 + 47/10,000 + 47/1,000,000 + ...

In Exercises 1–8, write the first five terms of each geometric sequence.

an = - 4a_(n-1), a1 = 10

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a(sub n) to find a(sub 8), the eighth term of the sequence. 1, 2, 4, 8, ...

In Exercises 51–56, the general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio.

an = n + 5

In Exercises 17–24, write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a7, the seventh term of the sequence.

18, 6, 2, 2/3, ...

In Exercises 51–56, the general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio.

an = n^2 + 5

Determining The Value of an Annuity

In Exercises 17–24, write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a7, the seventh term of the sequence.

1.5, - 3, 6, -12, ...

In Exercises 37–44, find the sum of each infinite geometric series.

3 + 3/4 + 3/4^2 + 3/4^3 + ...

Write a geometric sequence recursively

Exercises 88–90 will help you prepare for the material covered in the next section. Use the formula an = a₁3^(n-1) to find the seventh term of the sequence 11, 33, 99, 297,...

In Exercises 57–62, let
{a_n} = - 5, 10, - 20, 40, ..., 
{b_n} = 10, - 5, - 20, - 35, ...,
{c_n} = - 2, 1, - 1/2, 1/4

Find a10 + b10.

In Exercises 37–44, find the sum of each infinite geometric series.

1 + 1/3 + 1/9 + 1/27 + ...

In Exercises 37–44, find the sum of each infinite geometric series.

∞ Σ (i = 1) 8(- 0.3)^(i -- 1)

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30.

Find the sum of the first 14 terms of the geometric sequence: - 3/2, 3, - 6, 12, ...

In Exercises 63–64, find a2 and a3 for each geometric sequence.

8, a2, a3, 27

Evaluate the sum of a finite geometric series

In Exercises 1–8, write the first five terms of each geometric sequence.

a1 = 20, r = 1/2

A Quick Intro to Geometric Sequences

In Exercises 57–62, let
{a_n} = - 5, 10, - 20, 40, ..., 
{b_n} = 10, - 5, - 20, - 35, ...,
{c_n} = - 2, 1, - 1/2, 1/4

Find the difference between the sum of the first 10 terms of {an} and the sum of the first 10 terms of {bn}.

In Exercises 37–44, find the sum of each infinite geometric series.

1 - 1/2 + 1/4 - 1/8 + ...

Sum of a Geometric Series

How to determine the sum of a infinite geometric series

In Exercises 31–34, write the first five terms of each geometric sequence. a1 = 1/2, r = 1/2

Annuities: Annuity Due, Finding Future Value

In Exercises 31–36, find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. 

6 Σ (i = 1) (1/2)^(i + 1)<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/eb8a8506b806ff10-1678243841634.jpg" style="width: 126px;">

In Exercises 9–16, use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1 and common ratio, r.

Find a40 when a1 = 1000, r = - 1/2

Exercises 88–90 will help you prepare for the material covered in the next section. Consider the sequence 1, −2, 4, −8, 16, ………. Find a2/a3, a1/a2, a4/a3 and a5/a4 What do you observe?

In Exercises 9–16, use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1 and common ratio, r.

Find a12 when a1 = 5, r = - 2

Sum of an Infinite Geometric Series, Ex 2

Sum of an Infinite Geometric Series, Ex 3

In Exercises 17–24, write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a7, the seventh term of the sequence.

0.0004, - 0.0004, 0.04, - 0.04, ...

Finding The Sum of a Finite Geometric Series

Find the sum of each infinite geometric series. -6 + 4 - 8/3 + 16/9 - ...

In Exercises 63–64, find a2 and a3 for each geometric sequence.

2, a2, a3, - 54

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a(sub 6) when a(sub 1) = 16, r = 1/2

In Exercises 9–16, use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1 and common ratio, r.

Find a8 when a1 = 6, r = 2

In Exercises 1–8, write the first five terms of each geometric sequence.

an = - 5a_(n-1), a1 = - 6

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a(sub 5) when a(sub 1) = -3, r = 2

Sequences, Factorials, and Summation Notation

In Exercises 45–50, express each repeating decimal as a fraction in lowest terms.

0.257 ̅ (repeating 257)

Represent a geometric sequence as an explicit rule

In Exercises 51–56, the general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio.

an = 2^n

Express each repeating decimal as a fraction in lowest terms. 0.6 (repeating 6)

Sum of an Infinite Geometric Series, Ex 1

In Exercises 1–8, write the first five terms of each geometric sequence.

a1 = 5, r = 3

In Exercises 31–34, write the first five terms of each geometric sequence. a1 = 3, r = 2

In Exercises 31–36, find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. 

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In Exercises 9–16, use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1 and common ratio, r.

Find a8 when a1 = 1 000 000, r = 0.1

Finding the Sum of an Infinite Geometric Series

How to find the finite sum of a geometric sequence

In Exercises 31–36, find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. 

10 Σ (i = 1) 5 · 2^i<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/7f8205a86732d398-1678243812936.jpg" style="width: 97px;">

Use the formula for the sum of the first n terms of a geometric sequence to find the indicated sum. *sigma notation* i=1 start, 6 end, 5^i formula

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30.

Find the sum of the first 11 terms of the geometric sequence: 3, - 6, 12, - 24, ...

Geometric Sequences: A Formula for the' n - th ' Term , Part 1

Find the sum of the first 15 terms of the geometric sequence: 5, -15, 45, -135

Annuities: Annuity Due, Ex 2

Sum of Infinite Geometric Series

Find the sum of each infinite geometric series. 2 - 1 + 1/2 - 1/4 + ...

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1. Fundamental Concepts of Algebra

2. Equations and Inequalities

3. Functions and Graphs

4. Polynomial and Rational Functions

5. Exponential and Logarithmic Functions

6. Systems of Equations and Inequalities

7. Matrices and Determinants

8. Conic Sections

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Find the Common Difference for an Arithmetic Sequence

Write Terms of an Arithmetic Sequence

Use the Formula for the General Term of an Arithmetic Sequence

Use the Formula for the Sum of the First n Terms of an Arithmetic Sequence