Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 59. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | Geometric Sums 561 Calculating how many milligrams are in the body one week later requires simple addition. There are only eight terms so entering them into a calculator is not too much work. The second question how many milligrams will be in the body after several years of taking a mg pill every morning is certainly an important one but at flrst glance it looks like it will be a lot of work. Several years is not a well-deflned period of time but if we figured on about three years with a pill taken every day of the year we must sum 365 3 1095 terms 1096 terms if we look immediately after that 1096th pill is taken. The prospect of adding these terms up is not particularly appealing nor is the prospect of writing them down. But a mathematician would not become despondent at this prospect. Mathematicians look for patterns and there is a pattern to this sum. The challenge is how to exploit this pattern to cut down on the workload. After three years immediately after the 1096th pill is taken the number of milligrams of digitalis in the body is .04 .7 .04 .7 2 .04 .7 3 .04 .7 1095 .04 . We can use a precisely because there is a pattern. Each term from the second one on is times the previous term. The ratio of one term to the previous term is constant. Sums with this characteristic are called geometric sums and we will see that this characteristic makes it a real pleasure to add up a geometric sum. A finite geometric sum is a sum of the form 2 3 n a ar ar ar ar where a denotes the first term and r is the constant ratio of any term to the previous term. We ll begin by exploiting the pattern in order to find the amount of digitalis in the body immediately after the eighth pill. We use the strategy of giving a name to the thing we are trying to find. Let D .04 .7 .04 .7 2 .04 .7 3 .04 .7 4 .04 .7 5 .04 .7 6 .04 .7 7 .04 . Key Idea If we multiply D by we get something that looks very similar to D. .7D .7 .04 .7 2 .04 .7 3 .04 .7 4 .04 .7 5 .04 .7 6 .04 .7 7
