Problem 58 - 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. $$a+a r+a r^{2}+\dots+a r^{14}$$

Step-by-Step Solution

Answer

The sum of the sequence in summation notation is \(\sum_{k=0}^{14} a r^{k}\)

Step 1: Recognizing the Pattern

The numbers in the sequence have a common ratio of \(r\), and can be represented as multiples of \(a\) and increasing powers of \(r\).

Step 2: Compute the Formula of the nth Term

We observe that the nth term of sequence is given by \(a r^{n-1}\) where n is the position of the term in the sequence. Hence the first term is \(a\), the second term is \(ar\), the third term is \(ar^{2}\), and so forth. This gives us the formula of the nth term of the sequence.

Step 3: Expressing The Sequence in Summation Notation

We can now write the sequence in terms of summation notation. Using \(k\) as the index of summation and setting the lower limit of summation as 0 (our choice), the sequence can be represented as:\[\sum_{k=0}^{14} a r^{k}\]This sequences symbolizes adding up all terms of the sequence where in each term, the value of \(k\) goes from 0 to 14. The \(k^{th}\) term of the summation is \(a r^{k}\), which tallies with our observed pattern of the sequence.