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ss_4_6 - Section 4.6 Compound Interest The compound...

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Section 4.6. Compound Interest The compound interest formula is given by A n ( t ) = P (1 + r n ) nt where P = Principal (Present Value) A n ( t ) = Amount (Future Value) r = annual rate of interest (in decimal form) n = number of compounds per year t = number of years Example. $1000 is invested at a 6% annual rate. Find the amount in 2 years if the interest is compounded (a) yearly, (b) quarterly, (c) monthly, (d) daily. Solution. LetA = A n ( t ) = P (1 + r n ) nt = 1000(1 + . 06 n ) 2 n . Then (a) A = 1000(1 + . 06 1 ) 2 = 1 , 123 . 60 ( n = 1) (b) A = 1000(1 + . 06 4 ) 8 = 1 , 126 . 49 ( n = 4) (c) A = 1000(1 + . 06 12 ) 24 = 1 , 127 . 16 ( n = 12) (d) A = 1000(1 + . 06 365 ) 730 = 1 , 127 . 49 ( n = 365) Note. In the above example, A increases as n increases. What happens to A as n → ∞ ? The answer to this question is given by the following. A n ( t ) = P (1 + r n ) nt Pe rt as n → ∞ When A is given by A = Pe rt , the interest is said to be compounded continuously . Note: The continuous compounding formula shows how the number e 2 . 718 · · · occurs naturally. Example. Find the amount, A , after two years if $1,000 is invested at a nominal rate of 6% compounded continuously.
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