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An annuity pays 1 for the first n years, 2 for the second n years and 3 for the third n years with the effective annual interest rate $i$. Find the accumulated value of this annuity at time $3n$ directly and show that the accumulated value of this annuity is equivalent to $S3n|i + S2n|i + Sn|i$

For the first part got $Sn|i + S2n|i$

but I don't understand the second part.

Any ideas?

Nameless
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SpankyS
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1 Answers1

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It is clear that the accumulated value of the annuities can be presented by:

$S_{n|}(1+i)^{2n} + 2S_{n|}(1+i)^n+3S_{n|}$

We will now show that it simplifies to $S_{3n|}+S_{2n|}+S_{n|}$

$$S_{n|}(1+i)^{2n} + 2S_{n|}(1+i)^n+3S_{n|}$$
$$=\frac{(1+i)^n-1}i(1+i)^{2n} + 2\frac{(1+i)^n-1}i(1+i)^n+3\frac{(1+i)^n-1}i$$ $$=\frac{(1+i)^{3n}-(1+i)^{2n}}i+\frac{2(1+i)^{2n}-2(1+i)^{n}}i\frac{3(1+i)^{n}-3}i$$ Simplifying, this becomes: $$\frac{(1+i)^{3n}-1}i+\frac{(1+i)^{2n}-1}i+\frac{(1+i)^{n}-1}i$$ $$S_{3n|}+S_{2n|}+S_{n|}$$

As required.

user130512
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