a equation to demonstrate

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hero0765

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hi:
I have a mathematic equation 'A' to demonstrate:
1² + 3² + ...... + (2n+1)² = (n+1)(2n+1)(2n+3)/3

I have a known equation 'B':
1² + 2² + 3² + .... + n² = n(n+1)(2n+1) /6

how to use 'B' to demonstrate 'A' ?
 

Let
I(2n+1) = 1² + 3² + ...... + (2n+1)² = (n+1)(2n+1)(2n+3)/3
S = 1² + 2² + 3² + .... + n² = n(n+1)(2n+1) /6

But
S(2n+1) = 1² + 2² + 3² + .... + (2n+1)²
= 1² + 3² + .... + (2n+1)² + 2² + 4² + ...... (2n)²
= 1² + 3² + .... + (2n+1)² + 2²*( 1² + 2² + 3² + ...... n²)
= I(2n+1) + 4*S

==> I(2n+1) = S(2n+1) - 4*S
 

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    hero0765

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CMIIW :grin:

B = A + 2² + 4² + 6² + ... = A + 2²*A = 5*A

so A = B/5
 

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