1975 AMC 12 Problem 9

Attempt Problem 9 of the 1975 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1975 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

Let

a_1,

a_2,

\ldots

and

b_1,

b_2,

\ldots

be arithmetic progressions such that

a_1=25,

b_1=75,

and

a_{100}+b_{100}=100.

Find the sum of the first one hundred terms of the progression

a_1+b_1,

a_2+b_2,

\ldots.

0

100

10{,}000

505{,}000

not enough information given to solve the problem

The termwise sum of two arithmetic progressions is arithmetic

Its first and hundredth terms are both

100

Let

c_i=a_i+b_i.

This is an arithmetic progression with

c_1=25+75=100

and

c_{100}=100.

Equal endpoint terms force its common difference to be

0,

so all

100

terms equal

100.

Their sum is

10{,}000.

Therefore, the correct answer is C.

Problem 9 in other years