If ar is the coefficient of x10-r in the Binomial expansion of (1+x)10, then displaystyle∑r=110 r3((ar/ar-1))2 is equal to

Question

If ar is the coefficient of x10-r in the Binomial expansion of (1+x)10, then displaystyle∑r=110 r3((ar/ar-1))2 is equal to (A) 3025 (B) 1210 (C) 5445 (D) 4895

Tardigrade Admin · Accepted Answer

a r = 10 C 10- r = 10 C r ⇒ displaystyle∑ r =110 r 3(( 10 C r / 10 C r -1))2= displaystyle∑ r =110 r 3((11- r / r ))2= displaystyle∑ r =110 r (11- r )2 = displaystyle∑ r =110(121 r + r 3-22 r 2)=1210

Q. If $a_{r}$ is the coefficient of $x^{10 - r}$ in the Binomial expansion of $(1 + x)^{10}$ , then $r = 1 \sum 10 r^{3} (\frac{a _{r}}{a _{r - 1}})^{2}$ is equal to

3025

1210

5445

4895

$a_{r} =^{10} C_{10 - r} =^{10} C_{r}$
$\Rightarrow r = 1 \sum 10 r^{3} (\frac{^{10} C _{r}}{^{10} C _{r - 1}})^{2} = r = 1 \sum 10 r^{3} (\frac{11 - r}{r})^{2} = r = 1 \sum 10 r (11 - r)^{2}$
$= r = 1 \sum 10 (121 r + r^{3} - 22 r^{2}) = 1210$

Q. If ar​ is the coefficient of x10−r in the Binomial expansion of (1+x)10, then r=1∑10​r3(ar−1​ar​​)2 is equal to