If 7=5+1/7(5+α)+1/7²(5+2 α)+1/7³(5+3 α)+… … … … ∞, then the value of α is :

Solution

Maths sir · 4 steps

Split each bracket into a 5-part and an α-part: one plain GP and one standard AGP sum.

1. Write the general term. The nth term, counting from n = 0, is (5 + nα)/7ⁿ. So the right side is 5Σₙ₌₀^∞ 1/7ⁿ + α Σₙ₌₀^∞ n/7ⁿ.
2. Sum the GP. With x = 1/7, Σ xⁿ = 1/(1 - x) = 7/6. So the 5-part gives 5 × 7/6 = 35/6.

   Σₙ₌₀^∞ n xⁿ = x/(1-x)² = (1/7)/(36/49) = 7/36

3. Set up the equation. So 35/6 + 7α/36 = 7. Then 7α/36 = 7 - 35/6 = 7/6.
4. So. α = 7/6 × 36/7 = 6, which is option (d).

The idea to keep. For terms like (a + nd)xⁿ, split them into aΣ xⁿ and dΣ n xⁿ. Remember Σ n xⁿ = x/(1-x)² for |x| < 1. The x in the numerator matters.

Answer from the NTA final answer key. Solution by PrepWiser.