$10 \cdot 31$ Let $a_{1}, a_{2}, \cdots$ and $b_{1}, b_{2}, \cdots$ be arithmetic sequences, where $a_{1}=25, b_{1}$ $=75$ and $a_{100}+b_{100}=100$. The sum of the first 100 terms of the sequence $a_{1}+b_{1}, a_{2}+b_{2}, \cdots$ is 
(A) 0.
(B) 100.
(C) 10000.
(D) 505000.
(E) The given information is insufficient to solve the problem.
(26th American High School Mathematics Examination, 1975)
[Solution] The sum of two arithmetic sequences is still an arithmetic sequence. Let $d$ denote the common difference of the sequence $a_{1}+b_{1}, a_{2}+b_{2}, \cdots$. Then $99 d = (a_{100} + b_{100}) - (a_{1} + b_{1}) = 0$; thus $d = 0$. $\therefore \{a_{n} + b_{n}\}$ the sum of the first 100 terms is $100(a_{1} + b_{1}) = 10000$. Therefore, the answer is $(C)$.