Independent solution

How to solve this Increasing Perpetuity question

Setup

Setup

Let v = 1/1.105 and let n be the unknown number of payments. The present value supplied in the question is the value of an arithmetically increasing annuity-immediate.

i=0.105,v=1/1.105i=0.105,\quad v=1/1.105

Model

Model

The standard increasing-annuity identity converts the weighted payment sum to the displayed expression involving 1 − vⁿ. That form is monotone in n and therefore has a unique positive integer solution.

77.1=an‾∣i=1−vni277.1=\frac{a_{\overline n|}}{i}=\frac{1-v^n}{i^2}

Compute

Compute

Solving first for vⁿ gives 0.14997. Dividing logarithms yields n = 19.00, so no interpolation between payment counts is needed.

vn=0.14997,n=ln⁡(0.14997)ln⁡v=19v^n=0.14997,\quad n=\frac{\ln(0.14997)}{\ln v}=19

Answer

Answer

The calculation gives 19 for increasing perpetuity, matching published choice C.

n=19(C)\boxed{n=19\quad\text{(C)}}