If the coefficients of 2 nd , 3 rd and 4 th terms in the expansion of \((1+\text{x})^{\text{n}},…

If the coefficients of 2nd, 3rd and 4th terms in the expansion of \((1+\text{x})^{\text{n}}, \text{n}\in\text{N}\) are in A.P. then n = 

  1. None of these.

  2. 7

  3. 14

  4. 2

Solution

Solution:

Coefficients of 2nd, 3rd and 4th terms in the expansion of are \({^\text{n}}\text{C}_{\text{1}},{^\text{n}}\text{C}_{\text{2}}, {^\text{n}}\text{C}_{\text{3}}.\)

we have,

\(2\times{^\text{n}}\text{C}_{\text{2}}={^\text{n}}\text{C}_{\text{1}}+{^\text{n}}\text{C}_{\text{3}}\)

Dividing both sides by nC2, we get

\(2=\frac{{^\text{n}}\text{C}_{\text{1}}}{{^\text{n}}\text{C}_{\text{2}}}+\frac{{^\text{n}}\text{C}_{\text{3}}}{{^\text{n}}\text{C}_{\text{2}}}\)

\(\Rightarrow 2=\frac{2}{\text{n}-1}+\frac{\text{n}-2}{3}\)

\(\Rightarrow 6\text{n}-6=6+\text{n}^{2}+2-3\text{n}\)

\(\Rightarrow \text{n}^{2}-9\text{n}+14=0\)

\(\Rightarrow \text{n}=7\)

Asked in: RDSHARMA

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