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. *

#NumberTheory #MathematicalInduction #Series #Proofs #Sumofsquares

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
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Comments

We could use summation in these !

Mukul Sharma - 6 years, 1 month ago

I didn't get you.Plz elaborate

shivamani patil - 6 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$