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If 8 = 3 + 1 4 3 + p + 1 4 2 3 + 2 p + 1 4 3 3 + 3 p + … ∞ , then the value of p is

Correct answer

0

Step-by-step solution

Given that 3 + 1 4 3 + p + 1 4 2 3 + 2 p + … . to ∞ = 8 Let, S = 3 + 1 4 3 + p + 1 4 2 3 + 2 p + . . . ∞ ⇒ S = 3 + 3 4 + 3 4 2 + . . . + p 4 + 2 p 4 2 + . . . We know that, S ∞ = a + ar + ar 2 + . . . ∞ = a 1 - r Let, S 1 = 3 + 3 4 + 3 4 2 + . . . . ⇒ S 1 = 3 1 + 1 4 + 1 4 2 + . . . . ⇒ S 1 = 3 1 1 - 1 4 = 4 And, S 2 = p 4 + 2 p 4 2 + 3 p 4 3 + . . . . ⇒ S 2 4 = p 4 2 + 2 p 4 3 + 3 p 4 4 + . . . . ⇒ S 2 - S 2 4 = p 4 + p 4 2 + p 4 3 + . . . . . ⇒ 3 S 2 4 = p 1 4 + 1 4 2 + 1 4 3 + . . . . . ⇒ 3 S 2 4 = p 1 4 1 - 1 4

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