Combining Separate Summations
The image contains two separate problems, numbered 37 and 38. The instructions say, "Write each of 37-39 as a single summation." We'll focus on problem 37, as per your request.
Problem 37 is:
\[ \sum_{i=1}^{k} i^3 + (k + 1)^3 \]
We want to combine these two separate sums into one single summation. To do so, we can consider that the first summation is the sum of the cubes of the numbers from 1 to k, and the second term is the cube of (k+1), which does not depend on i and therefore is added k times (once for each value of i from 1 to k).
Thus, we can combine these terms into a single summation by extending the original sum from i=1 to k+1 for the cube of i.
The combined single summation is:
\[ \sum_{i=1}^{k+1} i^3 \]
The original summation adds the cube of each integer from 1 to k, and then we include the term (k + 1)^3 at the end, effectively summing the cubes of the integers from 1 to k+1.
