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

question 28

Essay

Use mathematical induction to prove the following for every positive integer $n$ .
$1 + 6 + 36 + 216 + \ldots + 6 ^ { n - 1 } = \frac { 6 ^ { n } - 1 } { 5 }$

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Definitions:

Methanol

A one-carbon alcohol, CH3OH, used as an industrial solvent, antifreeze, fuel, and as a precursor to other chemicals.

Trans-1-Iodo-4-Methylcyclohexane

A chemical compound featuring a cyclohexane ring with iodine and a methyl group attached in a trans configuration, indicating they are on opposite sides of the ring.

Rearrangement

A chemical reaction in which the structure of a molecule is rearranged to form a new molecule.

Expected Product

The outcome that is predicted to result from a chemical reaction under specified conditions, based on the reactants and the reaction mechanism.