kobel51 wrote: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

With evenly spaced integers:
Average = median = sum/number.
Sum = number*median.

In the problem at hand:
The median of the first five integers = sum/number = 560/5 = 112.
Thus, the 10 integers are as follows:
110, 111, 112, 113, 114
115, 116, 117...

The median of the last five integers is 117.
Thus:
Sum of the last five integers = number*median = 5*117 = 585.

The correct answer is A.

difference between any 1st and 6th consecutive integer is 5
difference between any 2nd and 7th consecutive integer is again 5
so the total difference will be 5*5=25
Ans is 560+25=585

kobel51 wrote: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

Solution:

In solving this problem we must first remember that when we have 10 consecutive integers we can display them in terms of just 1 variable. Thus, we have the following:

Integer 1: x
Integer 2: x + 1
Integer 3: x + 2
Integer 4: x + 3
Integer 5: x + 4
Integer 6: x + 5
Integer 7: x + 6
Integer 8: x + 7
Integer 9: x + 8
Integer 10: x + 9

We are given that the sum of the first 5 integers is 560. This means that:

x + x+1 + x+2 + x+3 + x+4 = 560

5x + 10 = 560

5x = 550

x = 110

The sum of the last 5 integers can be expressed and simplified as:

x+5 + x+6 + x+7 + x+8 + x+9 = 5x + 35

Substituting 110 for x yields:

(5)(110) + 35 = 585

Answer: A

Alternatively, because both equations have 5x in common, we know that the difference between the sum of the first five numbers and the sum of the last five numbers is the difference between (1+2+3+4) and (5+6+7+8+9). Since 35 - 10 = 25, the sum of the last 5 is 585, which is 25 more than 560.

Another option here: we've got a set of consecutive integers, so we know that the middle number = sum/(# of terms).

Given 5 integers with a sum of 560, the middle number is thus (560 / 5), or 112.

To get to the middle of the NEXT five integers, we just add 5. That means the middle number of the next five is 117, at which point we can work backwards: multiply this by 5, and we have a sum of 5*117, or 585.