Question

If
$\frac{1}{2\times 3 \times 4} \+ \frac{1}{3\times 4 \times 5 } + \frac{1}{4 \times 5 \times 6 }+ \dots + \frac{1}{100 \times 101 \times 102} = \frac{k}{101}$
then 34 k is equal to ____________.

Answer 1

The correct answer is 286
S=$$\frac{1}{2\times 3 \times 4} \+ \frac{1}{3\times 4 \times 5 } + \frac{1}{4 \times 5 \times 6 }+ \dots + \frac{1}{100 \times 101 \times 102}
$=\frac{1}{(3−1)⋅1}[\frac{1}{2×3}−\frac{1}{101×102}]$
$=\frac{1}{2}(\frac{1}{6}−\frac{1}{101×102})$
$=\frac{143}{102×101}=\frac{k}{101}$
∴ 34k = 286