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If T 2 / T 3 in the Expansion of ( a + B ) N and T 3 / T 4 in the Expansion of ( a + B ) N + 3 Are Equal, Then N = (A) 3 (B) 4 (C) 5 (D) 6 - Mathematics

MCQ

If  \[T_2 / T_3\]  in the expansion of \[\left( a + b \right)^n \text{ and } T_3 / T_4\]  in the expansion of \[\left( a + b \right)^{n + 3}\]  are equal, then n =

 
 

Options

  • 3

  •  4

  •  5

  •  6

     
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Solution

 5

\[\text{ In the expansion}  (a + b )^n , \text{ we have } \]

\[\frac{T_2}{T_3} = \frac{^{n}{}{C}_1 a^{n - 1} \times b^1}{^{n}{}{C}_2 a^{n - 2} \times b^2}\]

\[\text{ In the expansion } (a + b )^{n + 3} , \text{ we have } \]

\[\frac{T_3}{T_4} = \frac{^{n + 3}{}{C}_2 a^{n + 1} b^2}{^{n + 3}{}{C}_3 a^n b^3}\]

\[\text{ Thus, we have } \]

\[\frac{T_2}{T_3} = \frac{T_3}{T_4}\]

\[ \Rightarrow \frac{^{n}{}{C}_1 a}{^{n}{}{C}_2 b} = \frac{^{n + 3}{}{C}_2 a}{^{n + 3}{}{C}_3 b}\]

\[ \Rightarrow \frac{2}{n - 1} = \frac{3}{n + 1}\]

\[ \Rightarrow 2n + 2 = 3n - 3\]

\[ \Rightarrow n = 5\]

 
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 18 Binomial Theorem
Q 19 | Page 47
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