Question

The consecutive terms whose coefficients are equal in the expansion of $(5+10x{)}^{80}$ are

• A. $t_{27},t_{28}$
• B. $t_{53},t_{54}$
• C. $t_{54},t_{55}$
• D. $t_{26},t_{27}$

Answer 1

The correct option is C $t_{54},t_{55}$

$T_{r+1}={}^n{C}_r{a}^{n-r}{b}^r$
Applying to the above question we get
Coefficients of $T_{r+1}=T_{r+2}$
${}^{80}{C}_r{5}^{80-r}{10}^r={}^{80}{C}_{r+1}{5}^{80-r-1}{10}^{r+1}$
$\frac{80!.5}{(80-r)!\left(r\right)!}=\frac{80!.10}{(80-r-1)!(r+1)!}$
$\frac{5}{80-r}=\frac{10}{r+1}$
$5r+5=800-10r$
$15r=795$
$r=53$
Hence the terms are $T_{r+1}$ and $T_{r+2}$
Substituting the value of r we get the consecutive terms as
$t_{54}$, $t_{55}$