edited by
538 views
2 votes
2 votes

Answer is B as given in solution.

edited by

2 Answers

2 votes
2 votes

ANSWER: OPTION (B) $S_r = \frac{6.x}{(1-x)^4}$

Let, $S_r   =   3.2.1x   +   4.3.2x^2    +    5.4.3x^3    +    6.5.4x^4   + ….  $        

$S_r.x =  \ \ \ \ \ \\ \ \ \ \ \ \\ \ \ \ \ \\ \ \ \  3.2.1x^2   +   4.3.2x^3    +    5.4.3x^4 + …. $      {multiplying both sides with $x$}

-----------------------------------------------------------------------------------------------------------------------------------------

$S_r-S_rx=  6.x+     18.x^2     +    36.x^3   +    60.x^4 +….$ {Subtracting}

$S_r(1-x) =   6.x(1+3.x  +6.x^2  +10.x^3$

$S_r=  (6.x/1-x )  *   (1+3.x  +6.x^2  +10.x^3$

-----------------------------------------------------------------------------------------------------------------------------------------

$S_r = (\frac{6.x}{1-x}).(1+3x + 6.x^2+10.x^3+….)$ → $eqn(1)$

-----------------------------------------------------------------------------------------------------------------------------------------

Let $S1 = 1+3x + 6.x^2+10.x^3+….$

$S1.x =   \ \ \  \ \ \ \ \ \ 1.x+3.x^2 +6.x^3 + \ ...$

-----------------------------------------------------------------------------------------------------------------------------------------

$S1(1-x)= 1+2.x+3.x^2+4.x^3+...$ → $eqn(2)$ {Subtracting}

-----------------------------------------------------------------------------------------------------------------------------------------

Let $S2= 1+2.x+3.x^2+4.x^3+...$

$S2.x= \ \ \ \ \ \ \ \ \ 1.x+2.x^2+3.x^3+...$

-----------------------------------------------------------------------------------------------------------------------------------------

$S2(1-x)= 1+x+x^2+x^3+… = \frac{1}{1-x}$ → $eqn(3)$  {Subtracting}

-----------------------------------------------------------------------------------------------------------------------------------------

Hence, $S2 = \frac{1}{(1-x)^2}$

$\implies S1 = \frac{1}{(1-x)^3}$

Thus, $S_r = \frac{6.x}{1-x}. \frac{1}{(1-x)^3} \implies S_r = \frac{6.x}{(1-x)^4}$ 

 

Related questions

7 votes
7 votes
2 answers
1
0 votes
0 votes
1 answer
3
2 votes
2 votes
2 answers
4
Mayank Khakharia 1 asked Mar 29, 2018
1,769 views
What will be the coefficient of x^17 in the expansion of (x+x^2+x^3+x^4+x^5+x^6)^4?