SOLUTION: The sum of all of the integers from 1 to 30 is 465. The sum of all of the integers from 1 to 30 that are divisible by 3 is (A) 135 (B) 155 (C) 156 (D) 165 (E) None of these

Algebra ->  Divisibility and Prime Numbers -> SOLUTION: The sum of all of the integers from 1 to 30 is 465. The sum of all of the integers from 1 to 30 that are divisible by 3 is (A) 135 (B) 155 (C) 156 (D) 165 (E) None of these       Log On


   

Question 314469: The sum of all of the integers from 1 to 30 is 465. The sum of all of the integers from 1 to 30 that are divisible by 3 is
(A) 135 (B) 155 (C) 156 (D) 165 (E) None of these
Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!
The sum of all of the integers from 1 to 30 is 465. The sum of all of the integers from 1 to 30 that are divisible by 3 is 

3 + 6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 + 30 = 165, choice (D)

or do it by the formula for the sum of an arithmetic series,

with first term = a%5B1%5D=3, common difference d=+3, nth term = a%5Bn%5D=30

a%5Bn%5D+=+a%5B1%5D+%2B+%28n-1%29%2A3

30+=+3+%2B+%28n-1%29%2A3

27+=+%28n-1%29%2A3

27%2F3+=+%28%28n-1%29%2A3%29%2F3

9+=+n-1

10=n

Then use the sum formula:

S%5Bn%5D=%28n%2F2%29%28a%5B1%5D%2Ba%5Bn%5D%29

S%5B10%5D=%2810%2F2%29%283%2B30%29

S%5B10%5D=%285%29%2833%29

S%5B10%5D=165, choice (D)

Edwin