Geometric Sequence Question

Question

Find the 26th term of the geometric sequence with a5= 5/4 and a12 = 160.

Answer 1

Usually I write a geometric series starting with a₀ so that the a₁ = a₀r and a_n = a₀r^n-1.

a₁₂/a₇ = r⁷ = 128 so that r = 2

a₁₂ = a₀r¹¹ _{= a}0 *2¹¹ = 160 => a₀ = 5/32.

With that information you should be able to find a₂₆.

Answer 2

General form is a_n = a₁·r^n-1

To find r, the common ratio, take the ratio of the two given terms:

a₁₂ = a₁·r¹¹ = 160

a₅ = a₁·r⁴ = 5/4

a₁₂/a₅ = 160/(5/4) = a₁·r¹¹/a₁·r⁴

160/(5/4) = r⁷

128 = r⁷

2 = r

To find a₂₆, take its ratio with either of the two given terms

a₂₆/a₁₁ = a₁·2²⁵/a₁·2¹¹

a₂₆/160 = 2¹⁴

Solve for a₂₆