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 $n$ .

---

Definitions:

Planning Budget

A budget based on the level of output planned at the start of the budgetary period.

Spending Variance

A financial metric that compares the actual amount spent on a specific item to the budgeted or expected amount.

Cost Formulas

Equations or models used to calculate the costs associated with manufacturing a product or delivering a service.

Plane Operating Costs

Expenses related to the operation of an aircraft, including fuel, maintenance, crew salaries, and hangar fees.