The sum of sum of arithmetic progressions

Algebra Level 3

Consider an arithmetic progression with terms a 1 , a 2 , … , a n a_1, a_2, \ldots , a_n .

Define S m = a 1 + a 2 + ⋯ + a m S_m = a_1 + a_2 + \cdots + a_m and T m = S 1 + S 2 + ⋯ + S m T_m = S_1 + S_2 + \cdots + S_m .

Trivially, if we know the value of T 1 T_1 , we will also know the value of S 1 S_1 .

Without any other information given, does there exist a positive integer k > 1 k>1 such that if we know the value of T k T_k , then we will also know the value of S k S_k ?

Inspiration

Yes, infinitely many Yes, finitely many No

Pi Han Goh by Pi Han Goh , 30, Malaysia

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1 solution

We can uniquely determine S k S_{k} from the knowledge of T k T_{k} if m 3 \dfrac{m}{3} = m 2 \dfrac{m}{2} or m=0. Hence the answer is NO.

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