Problem 33 - College Algebra

Question

Find each indicated sum. $$\sum_{k=1}^{5} k(k+4)$$

Step-by-Step Solution

Answer

The sum of the series is 115.

Step 1: Understanding the Summation

The Sigma notation \( \sum_{k=1}^{5} k(k+4) \) represents a series where the variable \( k \) values start at 1 and end at 5. The function inside the Sigma, \( k(k+4) \), must be evaluated for each value of \( k \) between 1 and 5.

Step 2: Calculating Individual Terms

Evaluate the function at each of the \( k \) values. With \( k=1 \) it is \( 1(1+4)=5 \), with \( k=2 \) it is \( 2(2+4)=12 \), with \( k=3 \) it is \( 3(3+4)=21 \), with \( k=4 \) it is \( 4(4+4)=32 \), and with \( k=5 \) it is \( 5(5+4)=45 \).

Step 3: Adding the Terms

Add the results of each individual term to find the sum of the series, \( 5+12+21+32+45=115 \).