In an increasing, geometric series,


In an increasing, geometric series, the sum of the second and the sixth term is $\frac{25}{2}$ and the product of the third and fifth term is 25 . Then, the sum of $4^{\text {th }}, 6^{\text {th }}$ and $8^{\text {th }}$ terms is equal to :

  1. (1) 35

  2. (2) 30

  3. (3) 26

  4. (4) 32

Correct Option: 1


$a r+a r^{5}=\frac{25}{2}$

$a r^{2} \times a r^{4}=25$

$a^{2} r^{6}=25$

$a r^{3}=5$


$\frac{5 r}{r^{3}}+\frac{5 r^{5}}{r^{3}}=\frac{25}{2}$


Put $r^{2}=t$


$2 t^{2}-5 t+2=0$

$2 t^{2}-4 t-t+2=0$

$(2 t-1)(t-2)=0$

$t=\frac{1}{2}, 2 \Rightarrow r^{2}=\frac{1}{2}, 2$


$=a r^{3}+a r^{5}+a r^{7}$

$=a r^{3}\left(1+r^{2}+r^{4}\right)$



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