Use mathematical induction

Question

Use mathematical induction to prove that the statement is true for every positive integer n.

7+49+343+...+7ⁿ= 7ⁿ+1-7/6

Answer 1

USE PARENTHESIS!!!!

the actual formula is...

7 + 7^2 + 7^3 + .... + 7^(n) = [7^(n+1)-7]/6

n=1: [7^(1+1)-7 ] / 6 = [7^2-7]/6

= (49-7)/6 = 42/6 = 7 = 7^1

n=2: 7+7^2 = 7+49=56

vs

[ 7^3 - 7]/6 = 3(43-7)/6 = 336/6 = 56

induction hypothesis:

7 + 7^2 + 7^3 + .... + 7^(n) = [7^(n+1)-7]/6

Then

7 + 7^2 + 7^3 + .... + 7^(n) + 7^(n+1) =

[7^(n+1)-7]/6 + 7^(n+1) = <-- substitutes induction hypothesis

[7^(n+1)-7]/6 + 6*7^(n+1)/6 = <-- common denominator

[7^(n+1)-7 + 6*7*(n+1)]/6 = <-- combines numerators

[7*7^(n+1) - 7]/6 <--- combines like terms

[ 7^(n+2)-7]/6 <-- property of exponents