9. Sequences, Series, & Induction

Express each repeating decimal as a fraction in lowest terms. $0.\overline{257}$

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

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

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. a_n = 2ⁿ

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

Let {a_n} = - 5, 10, - 20, 40, ..., {b_n} = 10, - 5, - 20, - 35, ..., {c_n} = - 2, 1, - 1/2, 1/4 Find a₁₀ + b₁₀.

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 {a_n} and the sum of the first 10 terms of {b_n}.

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 6 terms of {a_n} and the sum of the infinite seris containing all the terms of {c_n}.

Find a₂ and a₃ for each geometric sequence. 8, a₂, a₃, 27

Find a₂ and a₃ for each geometric sequence. 2, a₂, a₃, - 54

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?

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?

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,...