Can a sequence of numbers that have a common difference of three ever sum to a power of 5?

Question

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Answer 1

No. It can never sum to a power of 5 if the power must be an integer. But if any rational or irrational power is allowed, it can sum to the power of any base

the Gauss formula for sum of a sequence with common difference of 1 is n(n+1)/2

to get the sum of a sequence with common difference of 3, the formula become 3n(n+1)/2

if that = 5^x where x is an integer, then n(n+1) = (2/3)5^x but 5^x will be an integer ending in 5, which can never be divisible by 3.