Problem 57 - College Algebra

Question

Express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. $$a+a r+a r^{2}+\dots+a r^{12}$$

Step-by-Step Solution

Answer

The given series can be expressed using the summation notation as: \(\sum_{k=0}^{12} a r^k\).

Step 1: Identify the pattern and the first term

The first term of the series is 'a' and each subsequent term is obtained by multiplying the previous term by the common ratio 'r'. This allows us to identify the pattern behind these terms. In the series, the powers of 'r' start from 0 and go till 12.

Step 2: Write the general term

Now, we need to write the general term which can be used to obtain all the terms in the series. The general term when 'r' is raised to the power 'k' and multiplied by 'a' is \(a r^k\).

Step 3: Write the series using the summation notation

Finally, write the entire series by using the summation notation. We are given that we can use any lower limit for the summation. Choosing lower limit as 0 (as the power of 'r' starts from 0) and upper limit will be 12 (as the power goes till 12), the series can be expressed as: \(\sum_{k=0}^{12} a r^k\)