I have been unable to solve the following exercise from my Probability and Statistics book.

I succeeded in expressing observations (ii) and (iii) as mathematical statements:Under an insurance policy, a maximum of five claims may be filed per year by a policyholder. Let $\displaystyle p_n$ be the probability that a policyholder files n claims during a given year, where $\displaystyle n$ = 1,2,3,4,5. An actuary makes the following observations:

(i) $\displaystyle p_n \geq p_{n+1}$ for $\displaystyle n$ = 1,2,3,4.

(ii) The difference between $\displaystyle p_n$ and $\displaystyle p_{n+1}$ is the same for $\displaystyle n$ = 1,2,3,4.

(iii) Exactly 40% of the policyholders file fewer than three claims during a given year.

Calculate the probability that a random policyholder will file more than three claims during a given year.

(A) 0.14 (B) 0.16 (C) 0.27 (D) 0.29 (E) 0.33

(ii) $\displaystyle p_n - p_{n+1} = k$ for $\displaystyle n$ = 1,2,3,4.

(iii) $\displaystyle p_1 + p_2 = 0.4$

I also know the probabilities sum to 1, so

$\displaystyle p_1 + p_2 + p_3 + p_4 + p_5 = 1$

To solve the exercise, I need to find $\displaystyle p_4 + p_5$, but I have been unable to do so. I tried rewriting the sum of probabilities like so:

$\displaystyle p_1 + (p_1 - k) + (p_1 - 2k) + (p_1 - 3k) + p_5 = 1 \iff 4p_1 - 6k + p_5 = 1$

I have tried other approaches, but it seems I am aimlessly solving equations that are just restatements of each other. I believe if I could find $\displaystyle p_1$ and $\displaystyle k$, then I could find $\displaystyle p_4$ and $\displaystyle p_5$. If anyone could point me in the right direction, I would appreciate it.

By the way, the solutions manual says the answer is (C) 0.27.