Use Mathematical Induction to Prove That the Formula $1 \cdot 3 + 2 \cdot 4 + 3 \cdot 5 + \cdots + n ( n + 2 ) = \frac { n ( n + 1 ) ( 2 n + 7 ) } { 6 }$

question 7

Essay

Use mathematical induction to prove that the formula $1 \cdot 3 + 2 \cdot 4 + 3 \cdot 5 + \cdots + n ( n + 2 ) = \frac { n ( n + 1 ) ( 2 n + 7 ) } { 6 }$ is true for all natural numbers $n$ .

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Definitions:

Marginal Returns

denotes the additional output obtained by adding one more unit of a specific input, while keeping other inputs constant.

Diseconomies of Scale

The scenario where costs start increasing as a company or project becomes too large, leading to inefficiency.

Bureaucratic

Relating to a system of government in which most of the important decisions are made by state officials rather than by elected representatives.

Large Management

Refers to the administration and execution of strategic objectives within large organizations, involving complex structures and substantial resource allocation.