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If the value of 1 + 2 3 + 6 3 2 + 10 3 3 + … . . upto ∞ log 0 . 25 1 3 + 1 3 2 + 1 3 3 + … . upto ∞ is l , then l 2 is equal to .

Correct answer

0

Step-by-step solution

l = 1 + 2 3 + 6 3 2 + 10 3 3 + … . .   upto   ∞ log 0 . 25 1 3 + 1 3 2 + 1 3 3 + … .   upto   ∞ Assume, S = 1 + 2 3 + 6 3 2 + 10 3 3 + … . .   upto   ∞ S = 1 + 2 3 + 6 3 2 + 10 3 3 + … - S 3 = 1 3 + 2 3 2 + 6 3 3 + … 2 S 3 = 1 + 1 3 + 4 3 2 + 4 3 3 + … 2 S 3 = 4 3 + 4 3 2 + 4 3 3 + … . S = 3 2 4 3 1 - 1 3 = 3 Now l = 3 log 0 . 25 1 3 1 - 1 3 l = 3 log 1 4 1 2 = 3 1 2 = 3 ⇒ l 2 = 3

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