Problem 32 - College Algebra

Question

Find each indicated sum. $$\sum_{i=1}^{5} i^{3}$$

Step-by-Step Solution

Answer

The sum of the cubes of the integers from 1 to 5 is 225.

Step 1: Identify the summation notation

Firstly, identify the summation notation. Here, it is denoted by \(\sum_{i=1}^{5} i^{3}\). This tells us to sum up, for \(i\) ranging from 1 to 5, the cube of \(i\). The 'i' under the summation symbol indicates the variable that will be changing and the 1 and 5 above and below are the starting and ending points respectively.

Step 2: Calculate the cube of each integer

Next, calculate the cube of each integer from 1 to 5, individually.\[1^{3} = 1 \]\[2^{3} = 8\]\[3^{3} = 27\]\[4^{3} = 64\]\[5^{3} = 125\]

Step 3: Compute the sum

The last step is to sum up all the calculated values.\[1 + 8 + 27 + 64 + 125 = 225\]