JEE PYQ: Sequence And Series Question 25
Question 25 - 2020 (03 Sep Shift 2)
If $m$ arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that $4^{th}$ A.M. is equal to $2^{nd}$ G.M., then $m$ is equal to _______.
Show Answer
Answer: (39)
Solution
Let $m$ arithmetic mean be $A_1, A_2, \ldots A_m$ and $G_1, G_2, G_3$ be geometric mean.
The A.P. formed by arithmetic mean is $3, A_1, A_2, A_3, \ldots, A_m, 243$
$\therefore d = \frac{243 - 3}{m+1} = \frac{240}{m+1}$
The G.P. formed by geometric mean: $3, G_1, G_2, G_3, 243$
$r = \left(\frac{243}{3}\right)^{1/4} = (81)^{1/4} = 3$
$\therefore A_4 = G_2$
$\Rightarrow 3 + 4\left(\frac{240}{m+1}\right) = 3(3)^2$
$\Rightarrow 3 + \frac{960}{m+1} = 27 \Rightarrow m+1 = 40 \Rightarrow m = 39$
BETA
