Use Mathematical Induction to Prove the Statement Is True for All

Essay

Use mathematical induction to prove the statement is true for all positive integers n.
- $$5 + 5 \cdot \frac { 1 } { 7 } + 5 \cdot \left( \frac { 1 } { 7 } \right) ^ { 2 } + \ldots + 5 \cdot \left( \frac { 1 } { 7 } \right) ^ { n - 1 } = \frac { 5 \left( 1 - \left( \frac { 1 } { 7 } \right) ^ { n } \right) } { 1 - \frac { 1 } { 7 } }$$