Multiple choice

The sum of first two terms of an infinite G.P. is $5$ and each term is three times the sum of following terms, then the 4th term of the series is $____.$

  1. $1$
  2. $\dfrac 14$
  3. $\dfrac 1{16}$
  4. $\dfrac 18$
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C Correct answer
Explanation

Let the G.P. be a, ar, ar^2... Sum of first two = a(1+r) = 5. Each term is 3 times the sum of following terms: a*r^n = 3 * (a*r^(n+1) / (1-r)). This simplifies to 1 = 3r / (1-r), so 1-r = 3r, 4r = 1, r = 1/4. Then a(1 + 1/4) = 5, a(5/4) = 5, a = 4. The 4th term is ar^3 = 4 * (1/4)^3 = 4 / 64 = 1/16.

AI explanation

Using the condition that each term is three times the sum of the following terms, we write a * r^(n-1) = 3 * (a * r^n) / (1 - r), which simplifies to r = 1/4. Given the sum of the first two terms is 5, we solve a + a/4 = 5 to get the first term a = 4. The 4th term is a * r^3 = 4 * (1/4)^3 = 4 * 1/64 = 1/16.

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