Question

Find the sum of 5 geometric means between $\frac{1}{3}$ and 243, by taking common ratio positive.

• A. 121
• B. 126
• C. 81
• D. 111

Answer 1

The correct option is A 121

Given that, there are $5$geometric means between the two numbers $\frac{1}{3}$and $243$ , we have to find $7=(5+2)$

terms in G.P. of which $\frac{1}{3}$ is the first, and$243$ the seventh. Let r be the common ratio;

then $243$ = the seventh term =$\left(\frac{1}{3}\right)r^{(7-1)}=\frac{1}{3}.r^6$.

Therefore,$r^6=3.x.243={\mathrm{3.3.3}}^4=3^6$;

whence $r=6$

and the series is$\frac{1}{3},1,3,9,27,81,243$

(using the standard form a, ar, ar², ar³ …… of a G.P. ).

Now, the geometric mean between two given quantities$a,b=\sqrt{ab}$

Therefore, the required geometric means are,

$\sqrt{\frac{1}{3}.x.3},\sqrt{1.x.9},\sqrt{3.x.27},\sqrt{9.x.82},\sqrt{27.x.243}$$

$=1,3,9,27,81$$

Therefore, the sum of the $5$ geometric means is

$1+3+9+27+81=121$

Hence, this is the answer.