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  • Answer please! - Binomial theorem and its simple applications - JEE Main

If \left ( 2+\frac{x}{3} \right )^{55}is expanded in the ascending powers of x and the coefficients of powers of x in two consecutive terms of the expansion are equal, then these terms are

  • Option 1)

    7th and 8th

  • Option 2)

    8th and 9th

  • Option 3)

    28th and 29th

  • Option 4)

    27th and 28th

 

Answers (1)

best_answer

As we have learned

General Term in the expansion of (x+a)^n -

T_{r+1}= ^{n}c_{r}\cdot x^{n-r}\cdot a^{r}
 

- wherein

Where r\geqslant 0 \, and \, r\leqslant n

r= 0,1,2,----n

 

 

 

T_{r+1}= ^{55}C_r(2)^{55-r}(\frac{x}{3})^r \: \: \&\\

T_{r}= ^{55}C_{r-1}(2)^{56-r}(\frac{x}{3})^{r-1}

NOw \: \: \: T_{r+1}= T_r \: \: for x= 1

=^{55}C_{r}(2)^{55-r}(\frac{x}{3})^{r}

= ^{55}C_{r-1}(2)^{56-r}(\frac{x}{3})^{r-1}\: \: \: for\: \: \: x = 1

Making only coefficient equal , 

\Rightarrow \frac{1}{r} \left ( \frac{1}{3} \right )\ \frac{1}{(56-r)}(2)

\Rightarrow 56- r = 6r \\ 7r = 56 \Rightarrow r = 8

Terms are 8 th and 9 th 

 

 

 

 


Option 1)

7th and 8th

Option 2)

8th and 9th

Option 3)

28th and 29th

Option 4)

27th and 28th

Posted by

gaurav

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