9. Sequences, Series, & Induction

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.${\sum }_{i=1}^6{\left(\frac{1}{2}\right)}^{i+1}$

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?

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

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, ...

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

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

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

Write the first five terms of each geometric sequence. a_n = - 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 a₅ when a₁ = -3, r = 2

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

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

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