Use mathematical induction to prove the following for every positive integer $n$.
$\sum _ { i = 1 } ^ { n } 36 i ^ { 5 } = 3 n ^ { 2 } ( n + 1 ) ^ { 2 } \left( 2 n ^ { 2 } + 2 n - 1 \right)$

Name: Use mathematical induction to prove the following for every positive integer $n$. $\sum _ { i = 1 } ^ { n } 36 i ^ { 5 } = 3 n ^ { 2 } ( n + 1 ) ^ { 2 } \left( 2 n ^ { 2 } + 2 n - 1 \right)$
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Question

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The Answer is i. Show true for

n = 1

\begin{aligned}36 \cdot 1 ^ { 5 } & = 3 \cdot 1 ^ { 2 } \cdot ( 1 + 1 ) ^ { 2 } \left( 2 \cdot 1 ^ { 2 } + 2 \cdot 1 - 1 ight) \36 & = 3 \cdot 4 \cdot 3 \36 & = 36 \1 & = 1\end{aligned}

ii. Assume true for

n = k

\sum _ { i = 1 } ^ { k } 36 i ^ { 5 } = 3 k ^ { 2 } ( k + 1 ) ^ { 2 } \left( 2 k ^ { 2 } + 2 k - 1 ight)

iii. Prove true for

n = k + 1

\begin{aligned}\sum _ { i = 1 } ^ { k + 1 } 36 i ^ { 5 } & = \sum _ { i = 1 } ^ { k } 36 i ^ { 5 } + 36 ( k + 1 ) ^ { 5 } \& = 3 k ^ { 2 } ( k + 1 ) ^ { 2 } \left( 2 k ^ { 2 } + 2 k - 1 ight) + 36 ( k + 1 ) ^ { 5 } \& = 3 ( k + 1 ) ^ { 2 } \left[ k ^ { 2 } \left( 2 k ^ { 2 } + 2 k - 1 ight) + 12 ( k + 1 ) ^ { 3 } ight] \& = 3 ( k + 1 ) ^ { 2 } \left[ 2 k ^ { 4 } + 14 k ^ { 3 } + 35 k ^ { 2 } + 36 k + 12 ight] \& = 3 ( k + 1 ) ^ { 2 } \left[ 2 k ^ { 4 } + 8 k ^ { 3 } + 8 k ^ { 2 } + 6 k ^ { 3 } + 24 k ^ { 2 } + 24 k + 3 k ^ { 2 } + 12 k + 12 ight] \& = 3 ( k + 1 ) ^ { 2 } \left[ 2 k ^ { 2 } \left( k ^ { 2 } + 4 k + 4 ight) + 6 k \left( k ^ { 2 } + 4 k + 4 ight) + 3 \left( k ^ { 2 } + 4 k + 4 ight) ight] \& = 3 ( k + 1 ) ^ { 2 } \left[ 2 k ^ { 2 } ( k + 2 ) ^ { 2 } + 6 k ( k + 2 ) ^ { 2 } + 3 ( k + 2 ) ^ { 2 } ight] \& = 3 ( k + 1 ) ^ { 2 } ( k + 2 ) ^ { 2 } \left[ 2 k ^ { 2 } + 6 k + 3 ight] \& = 3 ( k + 1 ) ^ { 2 } [ ( k + 1 ) + 1 ] ^ { 2 } \left[ 2 k ^ { 2 } + 4 k + 2 + 2 k + 2 - 1 ight] \& = 3 ( k + 1 ) ^ { 2 } [ ( k + 1 ) + 1 ] ^ { 2 } \left[ 2 \left( k ^ { 2 } + 2 k + 1 ight) + 2 ( k + 1 ) - 1 ight] \& = 3 ( k + 1 ) ^ { 2 } [ ( k + 1 ) + 1 ] ^ { 2 } \left[ 2 ( k + 1 ) ^ { 2 } + 2 ( k + 1 ) - 1 ight]\end{aligned}

Use Mathematical Induction to Prove the Following for Every Positive nnn

Use Mathematical Induction to Prove the Following for Every Positive $n$