If the 4 th term in the expansion of (ax + 1/x)ⁿ is 5/2, for all x ∈ R, then…
2019
If the 4 th term in the expansion of (ax + 1/x)ⁿ is 5/2, for all x ∈ R, then the values of a and n are:
- A.
1/2 , 6
- B.
1/2 , 3
- C.
1 , 3
- D.
cannot be found
Attempted by 9 students.
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Correct answer: A
Solution:
Write the general (k+1)th term of (a x + 1/x)^n: T_{k+1} = C(n,k) (a x)^{n-k} (1/x)^k = C(n,k) a^{n-k} x^{n-2k}.
The fourth term corresponds to k = 3, so its power of x is n - 2·3 = n - 6. For the fourth term to be the constant 5/2 for all real x, the exponent must be 0, hence n - 6 = 0 and n = 6.
Compute the coefficient when n = 6: the fourth term coefficient is C(6,3) a^{6-3} = C(6,3) a^3 = 20 a^3. Set this equal to 5/2:
20 a^3 = 5/2 ⇒ a^3 = 1/8 ⇒ a = 1/2 (taking the real cube root).
Conclusion: a = 1/2 and n = 6.
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