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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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