In a series of twenty consecutive integers, the sum of the first two integers is 37. What is the sum of the last three integers in the set?


A 107

B 108

C 109

D 110

E 111
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In a set of consecutive integers, the sum of the first two is 37

(1) First integer, \(x\)?
\(x + (x+1)=37\)
\(2x+1=37\)
\(x=18\)

(2) Last integer?
If \(x\) is the first integer, and there are 20 integers total, then there are 19 more
Add 19 to \(x=18\) (Consecutive = +1 for each integer in the set.)
\(18+19=37\)

If not sure, use a simpler example: There are 4 consecutive integers in the set.
\(x=18\). List them. 18, 19, 20, 21
There are 4 integers, but we need only +3 added to \(x=18\) to find the last integer.

20 integers in this set, but add only 19 to the first integer. \(18+19=37\)

(3) Sum of last three integers?

Last three integers: 37, 36, 35
Mentally add the units digits.
Units digit = 8
(Or, 37 + 36 + 35 = 108)

Answer B
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Letting x = the first number in the set, we can create the equation:

x + x + 1 = 37

2x = 36

x = 18

The first integer is 18, and the second integer is 19. Note that the position number of the integer + 17 = the value of the integer. We see that for the 1st integer (position 1), we have 1 + 17 = 18. For the 2nd integer (position 2), we have 2 + 17 = 19. Thus, the 18th integer in the set is (18 + 17) = 35. The 19th integer in the set is (19 + 17) = 36, and the 20th integer in the set is (20 + 17) = 37.

We want the sum of the 18th, 19th, and 20th integers in the set.
Thus, the sum of the last three integers is:

35 + 36 + 37 = 108

Answer: B
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