# expression for k, fm related question?

Here is the question, a loan of $10000 is to be repaid with annual payments at the end of each year for the next 20 years. First 5 years the payments are k per year, and second 5 years the payments are 2k per year and the third 5 years the payments are 3k per year and the last 5 years the payments are 4k per year. I need to find an expression for k.

Here I said:

k a angle 5 + 2k a angle 5*v^5 + 3k a angle 5* v^10 + 4k a angle 5* v^15 = 10000,

but the solution from the back of the book gives 10000 = 4k*a angle 20 - 3k*a angle 15 - 2k*a angle 10 - k*a angle 5,

my question is if my solution is wrong or not?

Thanks

### 1 Answer

- 9 years agoFavorite Answer
|---- k ----|---- 2k ----|---- 3k ----|---- 4k ----|-->

With the timeline as above, your solution seems correct.

10 000 = k a[5] + 2k a[5] * v^5 + 3k a[5] * v^10 + 4k a[5] * v^15 = k a[5] * (1 + 2 v^5 + 3 v^10 + 4 v^15) = k a[5] * Ia[4], where Ia[4] is the 4-year increasing annuity with j = i(1/5) / (1/5)

If you want to solve it as it shows at the back of your book,

10 000 = 4k a[20] - k a[15] - k a[10] - k a[5] = k ( 4 a[20] - a[15] - a[10] - a[5] )

|---- (4k - k - k - k) = k ----|---- (4k - k - k) = 2k ----|---- (4k - k) = 3k ----|---- 4k ----|-->

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