Q58 / 160 · TS EAMCET 2022 (19 Jul Shift 2) · Mathematics

The expansion of $(a+x)^n$ contains 15 terms. When $x=1$ the ratio of the neighbouring terms to the middle term in this expansion is 16 . Then the positive integral value of ' $a$ ' is

  • A
    1
  • B
    3
  • Correct answer
    C
    4
  • D
    2

Solution

Since, given expansion contains 15 terms hence $n=$ 14 for expansion $(1+a)^{14}$ $$ \begin{aligned} & \mathrm{T}_7=\mathrm{T}_{6+1}=14_{\mathrm{C}_6}(\mathrm{a})^8 \\ & \mathrm{~T}_{\mathrm{g}}=\mathrm{T}_{8+1}=14_{\mathrm{C}_8}(\mathrm{a})^6 \end{aligned} $$ Ratio $=\frac{14_{\mathrm{C}_6} \cdot \mathrm{a}^8}{14_{\mathrm{C}_6} \mathrm{a}^6}=\frac{14 !}{8 ! 6 !} \times \frac{8 ! \times 6 !}{14 !} \mathrm{a}^2$ $\Rightarrow \mathrm{a}^2=16$ or $\mathrm{a}= \pm 4$

Hence, the correct answer is option: 'C'.

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