I have this question,
Prove,
$7 + 77 + 777 +7777 + 77...$n digits..$77 = 7/81[(10^n × 10) - 9n - 10]$
By induction.
Now since this question was given in the exercise that involves proving various statements by strong induction, this one is to be done by using strong induction.
Although I did it using the "weak" induction, I'm not quite sure how can this be, if possibly, done by strong induction?
Here is my assumption that let $P(n) : 7+ 77 + 777 +7777 + 77...$n digits..$77$ $= 7/81[(10^n × 10) - 9n - 10]$
For base cases, (n is supposed to be natural number in question, by the way)
For $P(1)$, each side = $7$
For $P(2)$, each side = $84$
So, now assume, $P(n)$ is true for any $i$ such that, $0 ≤ i ≤ k$ or, for all values of $n$ upto $k$, $P(n)$ is true. Then we have to prove that $P(k+1)$ is true.
So, $7+ 77 + 777 +7777 + 77...$k+1 digits..$77$ $= 7/81[((10^k × 10)× 10) - 9(k + 1) - 10]$ Is what we have to prove.
What do I do next? How do I prove the above statement making use of my inductive hypothesis?