Hello, Trentt!

Determine if this geometric series converges.

If so, find the value to which it converges.

t1 = 81 and t5 = 1 (Two answers)

The first term is: .t1 .= .a .= .81 .[1]

The fifth term is: .t5 .= .ar^4 .= .1 .[2]

. . . . . . . . . . . . . ar^4 . . . . 1

Divide [2] by [1]: .------ .= . ---- . . → . . r^4 .= .1/81

. . . . . . . . . . . . . . a . . . . . 81

. . Hence: .r .= .±1/3

In both cases, |r| < 1 . . . Therefore, both series converge.

If r = 1/3, the sum is: .S .= .81/(1 - 1/3) .= .243/2

If r = -1/3, the sum is: .S .= .81/(1 + 1/3) .= .243/4