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Stitz-Zeager_College_Algebra_e-book

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Unformatted text preview: . Example 9.2.2. An ordinary annuity oﬀers a 6% annual interest rate, compounded monthly. 1. If monthly payments of \$50 are made, ﬁnd the value of the annuity in 30 years. 2. How many years will it take for the annuity to grow to \$100,000? Solution. 1. We have r = 0.06 and n = 12 so that i = A= r n = 0.06 12 = 0.005. With P = 50 and t = 30, 50 (1 + 0.005)(12)(30) − 1 ≈ 50225.75 0.005 Our ﬁnal answer is \$50,225.75. 9.2 Summation Notation 569 2. To ﬁnd how long it will take for the annuity to grow to \$100,000, we set A = 100000 and solve for t. We isolate the exponential and take the natural logarithm of both sides of the equation. 50 (1 + 0.005)12t − 1 0.005 12t − 1 10 = (1.005) 100000 = (1.005)12t = 11 ln (1.005)12t = ln(11) 12t ln(1.005) = ln(11) t= ln(11) 12 ln(1.005) ≈ 40.06 This means that it takes just over 40 years for the investment to grow to \$100,000. Comparing this with our answer to part 1, we see that in just 10 additional years, the value of the annuity nearly doubles. This is a l...
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