Question

If $S_n$ denotes the sun to n terms of the series $(1\le n\le 9)1+22+333+........+999999999,$ then for $n\ge 2$

• A. $S_n-S_{n-1}=\frac{1}{9}({10}^n-n^2+n)$
• B. $S_n=\frac{1}{9}({10}^n-n^2+2n-2)$
• C. $9(S_n-S_{n-1})=n({10}^n-1)$
• D. $S_3=356$

Answer 1

The correct option is C $9(S_n-S_{n-1})=n({10}^n-1)$

$a_1=1a_2=22=2+20=2\left(\right(\frac{1-{10}^2}{1-10}\left)\right)a_3=333=3+30+300=3\left(\right(\frac{1-{10}^3}{1-10}\left)\right)......a_n=n\left(\right(\frac{1-{10}^n}{1-10}\left)\right)\therefore S_n-S_{n-1}=a_n=n\left(\right(\frac{1-{10}^n}{1-10}\left)\right)or-9(S_n-S_{n-1})=n(1-{10}^n)or9(S_n-S_{n-1})=n({10}^n-1)$