Anna is on a progressive workout plan, so every day she adds 5% to her exercise time. If she starts by exercising 15 mins on the first day, how many minutes will she have exercised all together on day 45?
This is a geometric series, since the difference between the exercise time on any two days is greater than the difference between any prior two days. You could just add: and so on up to 45, but that would be horribly tedious. In this lesson, you will learn how to answer a question like this will little effort.
Sums of Finite Geometric Series
A finite geometric series is simply a geometric series with a specific number of terms. For example, consider the series: 50 + 25 + 12.5 + ....The series is geometric: the first term is 50, and the common ration is (1/2).
The sum of the first two terms is 50 + 25 = 75. We can write this as S2 = 75
The sum of the first three is 50 + 25 + 12.5 = 87.5. We can write this as S3 = 87.5
To find the value of Sn in general, we could simply add together the first n terms in a series. However, this would obviously be tedious for a large value of n. Given the regular pattern in a geometric series - every term is (1/r) of the previous term, and the nth term is an = a1rn - 1 , we can use induction to prove a formula for Sn .
The sum of the first n terms in a geometric series is
For example, for the series 50 + 25 + 12.5 + ... , the sum of the first 6 terms is:
The figure below shows the same calculation on a TI-83/4 calculator:
We can use this formula as long as the series in question is geometric.
Earlier, you were asked a question about Anna and her progressive workout plan.
Every day she adds 5% to her exercise time. If she starts by exercising 15 mins on the first day, how many minutes will she have exercised all together on day 45?
Use the formula:
Find the sum of the first 10 terms of a geometric series with a1 = 3 and r = 5.
The sum is 58,593.
Notice that because the common ratio in this series is 5, the terms get larger and larger. This means that for increasing values of n the sums will also get larger and larger. In contrast, in the series with common ratio (1/2), the terms gets smaller and smaller. This situation implies something important about the sum.
Find the sum of each series:
- The first term of a geometric series is 4, and the common ratio is 3. Find S8.
- The first term of a geometric series is 80, and the common ratio is (1/4). Find S7.
Prove the formula by induction.
Step 1) If n = 1, the nth sum is the first sum, or a1 . Using the hypothesized equation, we get . This establishes the base case.
Step 2) Assume that the sum of the first k terms in a geometric series is .
Step 3) Show that the sum of the first k+1 terms in a geometric series is .
|The sum is the sum, plus the term|
|Substitute from step 2, and substitute the term|
|The common denominator is|
|Simplify the fraction|
|It is proven.|
Therefore we have shown that for a geometric series. Now we can use this equation to find any sum of a geometric series.
Find the sum: (Hint: if an = 640 , what is n?).
Write the first 5 terms of the sequence: .
Just do the multiplication for each term
the first 5 terms are:
Write the 3rd, 4th, and 6th terms of: .
As with Example 6, just perform the operations on the indicated values of n:
the 3rd, 4th, and 6th terms are:
Find the sum of the series: .
We could calculate all of the values for and add them, getting:
Or we can use the formula:
Find the sum of the finite series. You may simply calculate the individual terms and add them, or you may use the formula: .
To see the Review answers, open this PDF file and look for section 7.9.