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Non-Inductive proof challenge.

Prove following results without induction.

1)\({ 1 }^{ 2 }+{ 2 }^{ 2 }+{ 3 }^{ 2 }+...+{ n }^{ 2 }=\frac { n\left( n+1 \right) \left( 2n+1 \right) }{ 6 } \)

2)\(1\cdot 2+2\cdot 3+3\cdot 4+....+n\left( n+1 \right) =\frac { n\left( n+1 \right) \left( n+2 \right) }{ 3 } \)

*Would like to see unique approaches.I have one. *

Note by Shivamani Patil
1 year, 8 months ago

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We could use summation in these ! Mukul Sharma · 1 year, 8 months ago

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@Mukul Sharma I didn't get you.Plz elaborate Shivamani Patil · 1 year, 8 months ago

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