Question

Calculate the sum of $7,\sqrt{42},6,....$

Answer 1

Number of terms $n=\infty$
Clearly given series is G.P
Common ratio $r=\frac{\sqrt{6}}{\sqrt{7}}$ , i .e $\left|r\right|<1$
First term $a=7$
Formula for sum of $n$ terms of G.P. is given by
$S_n=\frac{a(r^n-1)}{r-1}$
If $n=\infty$ & $\left|r\right|<1$ then formula for total sum is
$S=\frac{a}{1-r}$
Substituting the values, we get
$S=\frac{7}{1-\sqrt{\frac{6}{7}}}$
$S=\frac{7\sqrt{7}}{\sqrt{7}-\sqrt{6}}$
$S=\frac{7\sqrt{7}\times (\sqrt{7}+\sqrt{6})}{7-6}$
$S=49+7\sqrt{42}$