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Deducing a Formula for Compound Interest

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notes

Suppose `P_1` is the sum on which interest is compounded annually at a rate of R % per annum.
Let `P_1` = ` 5000 and R = 5. Then by the steps mentioned

1. `SI_1 = Rs.(5000 xx 5 xx 1)/100 `

so, `A_1 = Rs. 5000 + (5000 xx 5 xx 1)/100`

= `Rs.5000 (1+5/100)=P_2`                                 
    

Or `SI_1= Rs.(P_1 xx R xx 1 )/100`

Or `A_1=P_1 + SI_1 = P_1 + (P_1R)/100`

= `P_1(1+R/100) xx R/10 =P_2`

2. `SI_2 =Rs.5000(1+5/100)xx (5xx1)/100` 

`=Rs.(5000 xx 5)/100 (1+5/100)`  


`A_2 = rs.5000(1+5/100)+rs.(5000xx 5)/100 (1+5/100)`

`Rs.5000(1+5/100)(1+5/100)`                          

`Rs. 5000 (1+5/100)^2 = P_3`                            
                            

or  `SI_2 = (P_2xx Rxx1)/100`

`=P_1(1+R/100) xx R/100`

`=(P_1R)/100 (1+R/100)`

`A_2 = P_2 + SI_2`

`=P_1(1+R/100)+P_1R/100(1+R/100)`

`=P_1(1+R/100)(1+R/100)`

`=P_1(1+R/100)^2 = P_3`

The amount at the end of n years will be 
`A_n = P_1(1+R/100)^n`  Or we can say  
A = `P(1+R/100)^n`

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