Question Detail
Question
If the coefficients of the 2nd, 3rd and 4th terms in the expansion of (1 + x)^n are in A.P, then the value of n is
Correct Answer
Option B is the correct answer.
Detailed Explanation
Coefficients: 2nd term: C(n,1)=n; 3rd term: C(n,2)=n(n-1)/2; 4th term: C(n,3)=n(n-1)(n-2)/6. In AP: 2 × C(n,2) = C(n,1) + C(n,3). So 2 × n(n-1)/2 = n + n(n-1)(n-2)/6 ⇒ n(n-1) = n + n(n-1)(n-2)/6. Divide by n (n≠0): (n-1) = 1 + (n-1)(n-2)/6 ⇒ Multiply by 6: 6n-6 = 6 + (n-1)(n-2) ⇒ 6n-12 = (n-1)(n-2) = n²-3n+2 ⇒ 0 = n² - 9n + 14 ⇒ (n-2)(n-7)=0 ⇒ n=2 or n=7. n=2 gives only 3 terms, 2nd and 3rd coefficients are 2 and 1, 4th doesn't exist. So n=7.
Hint
Use the condition for AP: 2 × (middle term) = first + last.