Show that the amortization formula may be written as: R = P*i*(1 + i)^n/((1 + i)^n - 1)Here i is the interest charged per period, R the periodic...

The loan of a principal amount P has to be repaid with n
periodic payments of an amount R. The rate of interest is
i.

To determine R in terms of P, i and n we use the present
value formula. An amount X today would have a value equal to X*(1 + i)^t after t
periods. The same can be put the other way round as: an amount X paid after t periods is
equal to X/(1 + i)^t today. This is the present value of the payment that would be made
in the future. To repay the loan the sum of the present value of all the future payments
should be equal to P.

P = R/(1 + i) + R/(1 + i)^2 + R/(1 +
i)^3...+ R/(1 + i)^n ...(1)

The sum of n terms of a
geometric series a, ar, ar^2... is a*(1 - r^n)/(1 - r), for r <
1

In (1), we have the common ratio 1/(1 + i) < 1 and
the first term is R/(1 + i)

This gives [R/(1 + i)][(1 -
(1/(1 + i))^n]/(1 - 1/(1 + i)) = P

=> R[(1 - (1/(1 +
i))^n]/i = P

=> R((1 + i)^n - 1)/((1 + i)^n)*i =
P

=> R = P*i*(1 + i)^n/((1 + i)^n -
1)

The amortization formula is: R = P*i*(1 +
i)^n/((1 + i)^n - 1)