Question

If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309 , then the sum of its first nine terms is :

(1) 760

(2) 755

(3) 750

(4) 757 

Answer 1

Correct option is: (4) 757  

\( a r+a r^3+a r^5=21 \quad \cdots(1)\)   

\(a r^7+a r^9+a r^{11}=15309 \quad \cdots(2)\)

\( \frac{E q^n(2)}{E q^n(1)} \Rightarrow \frac{r^7\left(1+r^2+r^4\right)}{r\left(1+r^2+r^4\right)}=\frac{15309}{21} \)

\( r^6=\frac{15309}{21}=729 \)

\( r=3\)

Using (1) \(\quad ar \left(1+r^2+r^4\right)=21\)

\(r=3 \Rightarrow 3 a(1+9+81)=21\)

\(\mathrm{a}=\frac{7}{91}=\frac{1}{13}\)

Sum of first 9 term \(=\frac{\frac{1}{13}\left(3^9-1\right)}{3-1}\)

\(\Rightarrow \frac{1}{26}\left(3^9-1\right)=757\)