Need help with this old exam question please:
A number, N, consists of r digits chosen from 5, 6, 7, 8, 9, any ofwhich may be repeated any number of times. Let S be the sum of the digits in the number N and D be the sum of the digits in the number 2N. Proveby induction that for any positive integer r, that D=2S-9r.
I can see that the result is true for r=1 andr=2.
It is very hard to help if we have no idea what you have tried. Please explain what you are studying and what you have tried.
I have not worked out a full proof, but I'll explain how I would attack it initially.
The very first thing I would do is to develop a notation suitable for an inductive proof.
\(\displaystyle r\ is\ a\ positive\ integer.\)
\(\displaystyle i\ is\ a\ positive\ integer.\)
\(\displaystyle u_i\ is\ an\ integer\ such\ that\ 5 \le u_i \le 9.\)
\(\displaystyle \displaystyle N_r = \left(\sum_{i=1}^ru_i * 10^{(i-1)}\right).\)
\(\displaystyle \displaystyle S_r = \sum_{i=1}^ru_i.\)
So far, this has been purely mechanical, no real thought involved. Just constructing a foundation. But now I at least come to a place where I must think: what are the digits in 2N
r. I can't be sure what is right until I have completed a proof. I am still looking for a proof. But here is what I would try first.
\(\displaystyle Let\ v_i = u_i - 5 \implies 0 \le v_i \le 4\ and\ u_i = v_i + 5 \implies 1 \le (2v_i + 1) \le 9\ and\ 2u_i = 10 + 2v_i.\)
Now define the sum of the digits in 2N
r. See whether that leads to a proof. If not, come back and show where you got stuck.
EDIT: I have completed a proof. It does involve creating an expression for the sum of the digits of Nr and an expression for the sum of the digits of 2Nr. It is an ugly proof to put on an exam. It took me a while.