Sum of Sums of A.P.

Level 2

Given natural number N N and two real numbers A , B A,B where A ≠ B A \neq B

Consider all arithmetic progressions (A.P.) , with N N terms in ascending order , containing A , B A,B .

Find the sum of sums of all such A.P.

Give your answer for N = 100 , A = 34 , B = 55 N=100, A=34, B=55 .


The answer is 22027500.

Albert Yiyi by albert yiyi , 31, Malaysia

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1 solution

Albert Yiyi
Aug 18, 2019

hint: ascending = ascending + descending 2 \text{ascending} = \frac{\text{ascending} + \text{descending}}{2} ( 27 , 34 , 41 , 48 , 55 , 62 , 69 , 76 ) + ( 62 , 55 , 48 , 41 , 34 , 27 , 20 , 13 ) = ( 89 , 89 , 89 , 89 , 89 , 89 , 89 , 89 ) \ \ \ \ (27, {\color{#D61F06} 34}, 41, 48, {\color{#D61F06} 55}, 62, 69, 76) \\ + (62, {\color{#D61F06} 55}, 48, 41, {\color{#D61F06} 34}, 27, 20, 13) \\ = (89, 89, 89, 89, 89, 89, 89, 89)

ans: 1 4 N 2 ( N − 1 ) ( A + B ) \frac{1}{4}N^2 (N-1)(A+B)

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