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If the Coefficients of 2nd, 3rd and 4th Terms in the Expansion of ( 1 + X ) N , N ∈ N Are in A.P., Then N =(A) 7 (B) 14 (C) 2 (D) None of These - Mathematics

MCQ

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

  

Options

  • 7

  •  14

  • 2

  •  none of these

     
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Solution

 7

Coefficients of the 2nd, 3rd  and 4th terms in the given expansion are:

\[^{n}{}{C}_1 ,^{n}{}{C}_2 \text{ and }  ^{n}{}{C}_3 \]
\[\text{ We have } : \]
\[2 \times ^{n}{}{C}_2 = ^{n}{}{C}_1 +^{n}{}{C}_3 \]
\[\text{ Dividing both sides by } ^{n}{}{C}_2 , \text{ we get: }  \]
\[2 = \frac{^{n}{}{C}_1}{^{n}{}{C}_2} + \frac{^{n}{}{C}_3}{^{n}{}{C}_2}\]
\[ \Rightarrow 2 = \frac{2}{n - 1} + \frac{n - 2}{3}\]
\[ \Rightarrow 6n - 6 = 6 + n^2 + 2 - 3n\]
\[ \Rightarrow n^2 - 9n + 14 = 0\]
\[ \Rightarrow n = 7 \left( \because n \neq 2 \text{ as }  2 > 3 \text{ in the 4th term } \right)\]

 
Concept: Rth Term from End
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 18 Binomial Theorem
Q 28 | Page 48
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