# Deducing a Formula for Compound Interest

## Notes

### Deducing a Formula for Compound Interest:

Calculation of compound interest can be generalized.

Suppose P1 is the sum on which interest is compounded annually at a rate of R% per annum. Let P1 = Rs. 5000 and R = 5.

 1) SI1 = ₹ (5000 xx 5 xx 1)/100  so, A1 = ₹ 5000 + (5000 xx 5 xx 1)/100      = ₹ 5000 (1+5/100) = P2 SI1= ₹ ("P"_1 xx "R" xx 1 )/100 A1 = P1 + SI1 = P1 + ("P"_1"R")/100 = P1(1 + "R"/100) = P2 2) SI2 = ₹ 5000(1 + 5/100) xx (5xx1)/100  = ₹ (5000 xx 5)/100 (1+5/100) A2 = ₹ 5000(1 + 5/100) + ₹ (5000xx 5)/100 (1+5/100) = ₹ 5000(1+5/100)(1+5/100)   = ₹ 5000(1+5/100)^2 = P3 SI2 = ("P"_2 xx "R" xx 1)/100 = "P"_1(1 + "R"/100) xx "R"/100 = ("P"_1"R")/100 (1 + "R"/100) A2 = P2 + SI2 = "P"_1(1 + "R"/100) + "P"_1 "R"/100(1 + "R"/100) = "P"_1(1 + "R"/100)(1 + "R"/100) = "P"_1(1 + "R"/100)^2 = "P"_3

Proceeding in this way the amount at the end of n years will be

An = P1(1 + R/100)^n

A = P(1 + R/100)^n

## Example

Find CI on Rs. 12600 for 2 years at 10% per annum compounded annually.
We have,
A = P(1 + R/100)^n, where Principal (P) = ₹ 12600, Rate (R) = 10, Number of years (n) = 2
= ₹ 12600(1 + 10/100)^2
= ₹ 12600(11/10)^2
= ₹ 12600 xx 11/10 xx 11/10
= ₹ 15246.
CI = A - P
= ₹ 15246 - ₹ 12600
= ₹ 2646.
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