Problem Solving

swerve wrote: ↑

Wed Dec 02, 2020 10:18 am

In an increasing sequence of 10 consecutive integers, the sum of the first 5 integers is 560. What is the sum of the last 5 integers in the sequence?

A. 585
B. 580
C. 575
D. 570
E. 565

The OA is A

Source: Official Guide

Solution:

We can let x = 1st integer, and thus x + 4 = 5th integer. Since we have a consecutive set of integers, the average of the first five integers is equal to the median of the first five integers, which is x + 2. Thus, the sum of the first five integers is 5 * (x + 2) = 5x + 10. We are told that the sum of the first five integers is 560, so we obtain the following equation:

5x + 10 = 560

5x = 550

x = 110
The first integer is 110, so the 6th integer is 110 + 5 = 115 and the 10th integer is 110 + 9 = 119. Since the mean is equal to the median for the last five integers, the mean is 117. Thus, the sum of the last 5 integers is 117 * 5 = 585.

Alternate Solution:

In an increasing sequence of 10 consecutive integers, the 6th integer will be 5 more than the 1st integer, the 7th integer will be 5 more than the 2nd integer, and so on. We note that each of the last 5 integers is 5 more than its counterpart in the first 5 integers. Thus, the sum of the last 5 integers should be 5 * 5 = 25 more than the sum of the first 5 integers. Since we are given that the sum of the first 5 integers is 560, the sum of the last 5 integers is 560 + 25 = 585.

Answer: A