1975 AHSME Problems/Problem 9

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 hundred terms of the progression $a_1 + b_1, a_2 + b_2, \ldots$

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 100 \qquad \textbf{(C)}\ 10,000 \qquad \textbf{(D)}\ 505,000 \qquad \\ \textbf{(E)}\ \text{not enough information given to solve the problem}$

Solution

Notice that $a_{100}$ and $b_{100}$ are $25+99k_1$ and $75+99k_2$ , respectively.

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1975 AHSME Problems/Problem 9

Solution