If 12×3×4+13×4×5+14×5×6+…+1100×101×102=k101, then 34k is equal to _____.

12⋅3⋅4+13⋅4⋅5+……+1100⋅101⋅102=k101⇒4−22⋅3⋅4+5−33⋅4⋅5+…..+102−100100⋅101⋅102=2k101⇒12⋅3−13⋅4+13⋅4−14⋅5+…..+1100⋅101−1101⋅102=2k101⇒16−1101⋅102=2k101⇒2k=1016−1102⇒17×2k=17×1016−17102=17176−16=17166⇒34k=286

Book Online Demo

Check Your IQ

Try Test

Courses

Dropper NEET Course Dropper JEE Course Class - 12 NEET Course Class - 12 JEE Course Class - 11 NEET Course Class - 11 JEE Course Class - 10 Foundation NEET Course Class - 10 Foundation JEE Course Class - 10 CBSE Course Class - 9 Foundation NEET Course Class - 9 Foundation JEE Course Class -9 CBSE Course Class - 8 CBSE Course Class - 7 CBSE Course Class - 6 CBSE Course

Offline Centres

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 _____.

see full answer

Talk to JEE/NEET 2025 Toppers - Learn What Actually Works!

Real Strategies. Real People. Real Success Stories - Just 1 call away

An Intiative by Sri Chaitanya

answer is 286.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

$\frac{1}{2 \cdot 3 \cdot 4} + \frac{1}{3 \cdot 4 \cdot 5} + \dots \dots + \frac{1}{100 \cdot 101 \cdot 102} = \frac{k}{101}$
$\begin{array}{l} \Rightarrow \frac{4 - 2}{2 \cdot 3 \cdot 4} + \frac{5 - 3}{3 \cdot 4 \cdot 5} + \dots . . + \frac{102 - 100}{100 \cdot 101 \cdot 102} = \frac{2 k}{101} \\ \Rightarrow \frac{1}{2 \cdot 3} - \frac{1}{3 \cdot 4} + \frac{1}{3 \cdot 4} - \frac{1}{4 \cdot 5} + \dots . . + \frac{1}{100 \cdot 101} - \frac{1}{101 \cdot 102} = \frac{2 k}{101} \\ \Rightarrow \frac{1}{6} - \frac{1}{101 \cdot 102} = \frac{2 k}{101} \Rightarrow 2 k = \frac{101}{6} - \frac{1}{102} \end{array}$
$\begin{array}{l} \Rightarrow 17 \times 2 k = \frac{17 \times 101}{6} - \frac{17}{102} = \frac{1717}{6} - \frac{1}{6} = \frac{1716}{6} \\ \Rightarrow 34 k = 286 \end{array}$