How to prove this equation using mathematical induction? I need to prove the equation 1

Lovellss 2022-06-30 Answered
How to prove this equation using mathematical induction?
I need to prove the equation
1 3 = 1 + 3 + 5 + + ( 2 n 1 ) ( 2 n + 1 ) + ( 2 n + 3 ) + + ( 2 n + ( 2 n 1 ) ) using mathematical induction.
I tried solving this but I got stuck. I would be very thankful is someone could help me. Or maybe give me a references or hint.
My Attempt:
- For n = 1
1 3 = 1 2 n + 1
1 3 = 1 2 + 1
1 3 = 1 3 which is true.
- For n = k
1 3 = 1 + 3 + 5 + + ( 2 k 1 ) ( 2 k + 1 ) + ( 2 k + 3 ) + + ( 2 k + ( 2 k 1 ) )
- For n = k + 1
1 3 = 1 + 3 + 5 + + ( 2 k 1 ) + ( 2 k + 1 1 ) ( 2 k + 1 ) + ( 2 k + 3 ) + + ( 2 k + ( 2 k 1 ) )
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Answers (1)

Punktatsp
Answered 2022-07-01 Author has 22 answers
Step 1
You did some mistake in the expression for the case n = k + 1 in the induction step. Morover it is convenient proceed as follows using that A B = 1 3 3 A = B.
We need to show by induction that that for any n 1.
1 3 = i = 1 n 1 + 3 + 5 + + ( 2 i 1 ) ( 2 i + 1 ) + ( 2 i + 3 ) + + ( 2 i + ( 2 i 1 ) )
- base case: n = 1 1 3 = 1 2 + 1
- induction step we assume
1 3 = i = 1 n 1 + 3 + 5 + + ( 2 i 1 ) ( 2 i + 1 ) + ( 2 i + 3 ) + + ( 2 i + ( 2 i 1 ) )
3 i = 1 n ( 1 + 3 + 5 + + ( 2 i 1 ) ) = i = 1 n ( 2 i + 1 ) + ( 2 i + 3 ) + + ( 2 i + ( 2 i 1 ) )
Step 2
Then 3 i = 1 n + 1 ( 1 + 3 + 5 + + ( 2 i 1 ) ) = i = 1 n + 1 ( 2 i + 1 ) + ( 2 i + 3 ) + + ( 2 i + ( 2 i 1 ) )
3 ( 2 ( n + 1 ) 1 ) = 2 n + 2 ( n + 1 ) + ( 2 ( n + 1 ) 1 )
3 ( 2 n + 1 ) = 2 n + 2 n + 2 + ( 2 n + 1 )
6 n + 3 = 4 n + 2 + ( 2 n + 1 )
6 n + 3 = 6 n + 3

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