346 CHAPTER 5 Exponential and Logarithmic Functions The new principal is + = + = P I $1000 $5 $1005. At the end of the second quarter, the interest on this principal is = ⋅ ⋅ = I $1005 0.02 1 4 $5.03 At the end of the third quarter, the interest on the new principal of + = $1005 $5.03 $1010.03 is = ⋅ ⋅ = I $1010.03 0.02 1 4 $5.05 Finally, after the fourth quarter, the interest is = ⋅ ⋅ = I $1015.08 0.02 1 4 $5.08 After 1 year the account contains + = $1015.08 $5.08 $1020.16. The pattern of the calculations performed in Example 1 leads to a general formula for compound interest. For this purpose, let P represent the principal to be invested at a per annum interest rate r that is compounded n times per year, so the time of each compounding period is n 1 year. (For computing purposes, r is expressed as a decimal.) The interest earned after each compounding period is given by formula (1). = ⋅ ⋅ = ⋅ ⋅ = ⋅ P r n P r n Interest principal rate time 1 The amount A after one compounding period is ( ) = + ⋅ = ⋅ + A P P r n P r n 1 After two compounding periods, the amount A, based on the new principal ( ) ⋅ + P r n 1 , i s ( ) ( ) ( ) ( ) ( ) = ⋅ + + ⋅ + ⋅ = ⋅ + ⋅ + = ⋅ + A P r n P r n r n P r n r n P r n 1 1 1 1 1 2 New principal Interest on new principal After three compounding periods, the amount A is ( ) ( ) ( ) ( ) ( ) = ⋅ + + ⋅ + ⋅ = ⋅ + ⋅ + = ⋅ + A P r n P r n r n P r n r n P r n 1 1 1 1 1 2 2 2 3 Continuing this way, after n compounding periods (1 year), the amount A is ( ) = ⋅ + A P r n 1 n Because t years will contain ⋅ n t compounding periods, the amount after t years is ( ) = ⋅ + A P r n 1 nt THEOREM Compound Interest Formula The amount A after t years due to a principal P invested at an annual interest rate r, expressed as a decimal, compounded n times per year is ( ) = ⋅ + A P r n 1 nt (2) In equation (2), the amount A is typically referred to as the accumulated value of the account, and P is called the present value . P r n Factor out · 1( ) ↑ +
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