JEE Main202229 Jul 2022Morning ShiftMathematicsSequences and SeriesActual
If 1 2 × 3 × 4 + 1 3 × 4 × 5 + 1 4 × 5 × 6 + … + 1 100 × 101 × 102 = k 101 , then 34 k is equal to _______.
Correct answer
0
Step-by-step solution
Given, 1 2 . 3 . 4 + 1 3 . 4 . 5 + … . + 1 100 . 101 . 102 = k 101 ⇒ 2 2 . 3 . 4 + 2 3 . 4 . 5 + … . + 2 100 . 101 . 102 = 2 k 101 ⇒ 4 - 2 2 . 3 . 4 + 5 - 3 3 . 4 . 5 + … . . + 102 - 100 100 . 101 . 102 = 2 k 101 ⇒ 1 2 × 3 - 1 3 × 4 + 1 3 × 4 - 1 4 × 5 + … . + 1 100 × 101 - 1 101 × 102 = 2 k 101 ⇒ 1 2 × 3 - 1 101 × 102 = 2 k 101 ⇒ 101 2 × 3 - 101 101 × 102 = 2 k ⇒ 2 k = 101 6 - 1 102 ⇒ 34 k = 286