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# Manipulation of a summation with application to finance math and interest rates.

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In financial mathematics, when valuing cash flows, an important function is the so-called immediate annuity-certain, a_n, which is defined by the relation

a_n = v + v^2 + v^3 + ....... + v^n

where v:=(1+i)^(-1) and i represents the effective rate of interest per annum. Prove that

a_n = (1-v^n)/i

If the effective rate of interest is 3% pa, find a_4 to 3 decimal places.

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#### Solution Preview

By definition

a_n = v + v^2 + .... + v^n (1)

and so multiplying both sides of (1) by v gives

va_n = v^2 + v^3 + .... + v^{n+1}. ...

#### Solution Summary

A proof that relates annuity-certain to interest rate is performed.

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