Question

Find the two consecutive terms in the expansion of $(3+2x{)}^{74}$ so that the coefficients of power of x are equal.

• A. $29\text{and}30$
• B. $30\text{and}31$
  
• C. $31\text{and}32$
• D. $28\text{and}29$

Answer 1

The correct option is B $30\text{and}31$

$T_{r+1}{=}^{74}{C}_r{3}^{74-r}\left(2x{)}^{r}andthen{T}_{r+2}{=}^{74}{C}_{r+1}{3}^{73-r}\right(2x{)}^{r+1}asco-efficientsareequal{}^{74}{C}_r{3}^{73-r}{2}^r{=}^{74}{C}_{r+1}{3}^{73-r}{2}^{r+1}(\frac{74!}{74-r}!r!)\times =\left(\frac{74!}{(73-r)!(r+1)!}\right)\times 23(73-r)!(r+1)!=2(74-r)!r!=3(73-r)!(r+1)!=2(74-r)!(73-r)!r!=3(r+1)=2(74-r)=3r+3=148-2r\therefore 5r=148-3r=\left(\frac{145}{5}\right)=29Hencetermsare(29+1{)}^{th}and(29+2{)}^{nd}or{30}^{th}and{31}^{th}term$