Case: <munderover> &#x2211;<!-- ∑ --> <mrow class="MJX-TeXAtom-ORD"> n =

Loreenwk

Loreenwk

Answered question

2022-05-22

Case:
n = 1 ( 1 ) n + 1 ( 2 n 1 )

Answer & Explanation

Fahrleine9m

Fahrleine9m

Beginner2022-05-23Added 11 answers

This can be easily proved by induction. The formula can be guessed by looking at the partial sums:
S 1 = 1 , S 2 = 2 , S 3 = 3 , S 4 = 4 , . .
It's easy to see that S 1 = ( 1 ) n + 1 n | n = 1 = 1 , so the formula works for the base case. Now assume the formula holds for arbitrary n, i.e. S n = ( 1 ) n + 1 n. The we have
S n + 1 = S n + ( 1 ) n + 2 ( 2 ( n + 1 ) 1 ) = ( 1 ) n + 1 n + ( 1 ) n + 2 ( 2 n + 1 ) = ( 1 ) n + 1 ( n 2 n 1 ) = ( 1 ) n + 1 ( n 1 ) = ( 1 ) n + 2 ( n + 1 )
which proves the result!

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