In an increasing geometric progression of positive terms, the sum of the second and sixth terms is 70 3 and the product of the third and fifth terms…

$\mathrm{T}_2+\mathrm{T}_6=\frac{70}{3}$T2​+T6​=370​ $\mathrm{ar}+\mathrm{ar}^5=\frac{70}{3}$ar+ar5=370​ $\mathrm{~T}_3 \cdot \mathrm{T}_5=49$ T3​⋅T5​=49 $\mathrm{ar}^2 \cdot \mathrm{ar}^4=49$ar2⋅ar4=49 $\mathrm{a}^2 \mathrm{r}^6=49$a2r6=49 $\mathrm{ar}^3=+7, \mathrm{a}=\frac{7}{\mathrm{r}^3}$ar3=+7,a=r37​ $\mathrm{ar}\left(1+\mathrm{r}^4\right)=\frac{70}{3}$ar(1+r4)=370​ $\frac{7}{\mathrm{r}^2}\left(1+\mathrm{r}^4\right)=\frac{70}{3}, \mathrm{r}^2=\mathrm{t}$r27​(1+r4)=370​,r2=t $\frac{1}{\mathrm{t}}\left(1+\mathrm{t}^2\right)=\frac{10}{3}$t1​(1+t2)=310​ $3 \mathrm{t}^2-10 \mathrm{t}+3=0$3t2−10t+3=0 $\mathrm{t}=3, \frac{1}{3}$t=3,31​ Increasing G.P. $\mathrm{r}^2=3, \mathrm{r}=\sqrt{3}$r2=3,r=3​ $\mathrm{T}_4+\mathrm{T}_6+\mathrm{T}_8$T4​+T6​+T8​ $=\mathrm{ar}^3+\mathrm{ar}^5+\mathrm{ar}^7$=ar3+ar5+ar7 $=\operatorname{ar}^3\left(1+\mathrm{r}^2+\mathrm{r}^4\right)$=ar3(1+r2+r4) $=7(1+3+9)=91$=7(1+3+9)=91