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Deducing a Formula for Compound Interest

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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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