Question

Let $\sum _{r=m}^{n-1}\frac{1}{r(r+1)}=\frac{1}{11}(wherem,nϵN)$ then

• A. 5n is divisible by 11
• B. m is divisible by 10
• C. mn = 1100
• D. ordered pair (m, n) is unique

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A m is divisible by 10
B mn = 1100
C 5n is divisible by 11
D ordered pair (m, n) is unique

$\sum _{r=m}^{n-1}\left(\frac{1}{r(r+1)}\right)\sum _{r=m}^{n-1}(\frac{1}{r}-\frac{1}{r+1})=\frac{1}{m}-\frac{1}{n}=11$
$\Rightarrow n=\frac{11m}{11-m}\Rightarrow n=\frac{11(m-11+11)}{11-m}$
$=\frac{121}{11-m}=-11$
Hence $11-m$ is a divisor of $121,$ so $11-m=1$
$\Rightarrow m=10$
Hence A, B, C, D are correct.