9. Sequences, Series, & Induction

Exercises 88–90 will help you prepare for the material covered in the next section. Consider the sequence whose nth term is a_n = (3)5ⁿ 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, ...

Express each repeating decimal as a fraction in lowest terms. $0.\bar{5}=\frac{5}{10}+\frac{5}{100}+\frac{5}{1000}+\frac{5}{10,000}+\cdots$

Find the $10^{\(\th\)}$ term of the geometric sequence in which $a_1=5$ and $r=2$.

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a₇ when a₁ = 2, r = 3

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, ...

Express each repeating decimal as a fraction in lowest terms. $0.\overline{47}=\frac{47}{100}+\frac{47}{10,000}+\frac{47}{1,000,000}+\cdots$

Write the first five terms of each geometric sequence. a_n = - 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, ...

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. a_n = n + 5

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, ...

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. a_n = n² + 5

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, ...

Find the sum of each infinite geometric series. 3 + 3/4 + 3/4² + 3/4³ + ...