Damon Cowan

Damon Cowan

Answered

2022-09-14

Prove n 1 n + n 1 n n 3 n 2 + n 1 n n 3 n 2 n 5 n 4 + . . . = n 1 3 without induction
I found this identity using Maple. Is there a (simple) way to prove it without using induction? Using induction, the proof is quite easy.
Prove for odd n that
k = 1 ( n + 1 ) / 2 j = 0 k 1 ( n 2 j 1 n 2 j ) = n 1 n + n 1 n n 3 n 2 + n 1 n n 3 n 2 n 5 n 4 + . . . = n 1 3

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Answer & Explanation

Skye Hamilton

Skye Hamilton

Expert

2022-09-15Added 14 answers

We need to prove that
n 1 n + n 1 n n 3 n 2 + n 1 n n 3 n 2 n 5 n 4 + . . . = n 1 3
or
1 + n 3 n 2 + n 3 n 2 n 5 n 4 + . . . = n 3
or
n 3 n 2 + n 3 n 2 n 5 n 4 + . . . = n 3 3
or
1 + n 5 n 4 + n 5 n 4 n 7 n 6 . . . = n 2 3
or
1 + 2 3 = 5 3 .
Done!

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