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

question 22

Essay

Use mathematical induction to prove the following for every positive integer $n$ .
$$\sum _ { i = 1 } ^ { n } 96 i ^ { 5 } = 8 n ^ { 2 } ( n + 1 ) ^ { 2 } \left( 2 n ^ { 2 } + 2 n - 1 \right)$$

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Definitions:

Discretionary Income

The amount of an individual's income left for spending, investing, or saving after all taxes and necessities have been paid.

Luxury Items

High-end products or services that are not essential but are desired for their quality, status, and aesthetic appeal, often associated with a higher price.

Voyage

A long journey involving travel, especially by sea or in space.

Applied Science

A discipline that applies existing scientific knowledge to develop more practical applications, including technology or inventions, within fields such as engineering and medicine.