How many digits are there before the hundredth $9$ in the following number: $97977977797777977777\cdots$

Question

When I count from left of the following number, how many digits are there before the hundredth $9$: $$97977977797777977777\cdots$$

Before the 3rd $9$ there are $2$ sevens and before the 5th nine there are $4$ sevens... So basically there are $99$ sevens before the hundredth nine + $99$ nines + the other sevens...

So how do I calculate the total number of the other sevens here? Am I right till what I have done?

Before the first nine: $0$ sevens. Before the second nine: $1$ seven. Before the third nine $1+2$ sevens. Before the fourth nine: $1+2+3$ sevens. Before the fifth nine: $1+2+3+4$ sevens. Can you go on? — Crostul, May 09 '17 at 15:23
Before the hundredth nine it would be 1+2+3+4.......+99=4950 nines?@Crostul — Justuraveragemathsstudent, May 09 '17 at 15:26
@Justuraveragemathstudent: Do not worry, it's only important that you thought about the problem and wrote down what your thoughts and attempt were. It seems you did, so you shouldn't be sorry. — Vincenzo Oliva, May 09 '17 at 16:09
@VincenzoOliva, I agree. "Easy" is a relative term anyway. If you don't know how to solve a problem, then that problem is not easy, for you, even if it is for others. — Barry Cipra, May 09 '17 at 16:47
@VincenzoOliva thx !! But let me just make sure...beginners can use this site too right? — Justuraveragemathsstudent, May 10 '17 at 10:56

Barry Cipra · Answer 1 · 2017-05-09T16:04:19.727

1

If you include the $n$th $9$, the answer would be

$$1+2+\cdots+n={n(n+1)\over 2}$$

By not including it, the answer is

$${n(n+1)\over2}-1={n^2+n-2\over2}={(n-1)(n+2)\over2}$$

edited May 09 '17 at 16:04

answered May 09 '17 at 15:28

Barry Cipra

79,832

That would make it (100)(101)/2=5050.But if u calculate the number of sevens its 4950 plus the 99 nines gives 5049 right? – Justuraveragemathsstudent May 09 '17 at 15:31
@Justuraveragemathsstudent, sorry, I made a sign error in my second step. They should agree now. Thanks for checking! – Barry Cipra May 09 '17 at 16:05

How many digits are there before the hundredth $9$ in the following number: $97977977797777977777\cdots$

1 Answers1