Question

John has taken out a loan for college. He started paying off the loan with a first payment of $100. Each month he pays, he wants to pay back 1.1 times as the amount he paid the month before. Explain to John how to represent his first 20 payments in sigma notation. Then explain how to find the sum of his first 20 payments, using complete sentences. Explain why this series is convergent or divergent. (10 points)

Answer 1

Sigma notation: 20 on top, n=1 on bottom and 100(1.1)^n-1 on the right of the sigma symbol, sum: 5727.4999, divergent

Explanation:

To represent something is sigma notation you start with the sigma symbol in the center. On the top, you put the last term in the series, or the upper limit, which would be 20 in this case. On the bottom, you put the first term a.k.a. lower limit which is 1 in this case. On the left, you put the expression to find the value of the nth term which is 100(1.1)n-1 since it is a geometric sequence. To find the sum of a geometric sequence you can use the formula (a1-a1r^n)/(1-r). a1 is 100 and r, the common ratio is 1.1, and n is 20. We can plug all these values in to get (100 - 100(1.1)20 )/(1-1.1) = (100 - 100(6.7275)/(-0.1) = (100 - 672.7499)/(-0.1) = 5727.4999. This series is divergent because the absolute value of r is great than 1.

Answer 2

The infinite sequence of geometric terms is divergent if you try to add up all the terms. Sure the first 20 terms will add up to a fixed number but this isn't true for an infinite number of terms. The reason why the infinite series diverges is because r = 1.1 is larger than 1. If r > 1 then the infinite series diverges. It only converges if -1 < r < 1.

To write this in sigma notation, you would write
`\displaystyle \sum_(n = 1)^(20)100(1.1)^(n-1)` which is the result of adding the terms of 100(1.1)^(n-1) for n = 1 all the way up to n = 20. You can compute this by hand or preferably with a calculator or spreadsheet program