Find the sum of $\;$ $2 + 6 + 18 + 54 + \cdots + 4374$
$2, \; 6, \; 18, \; 54, \; \cdots, \; 4374$ is a geometric progression (G.P) with
first term $= a = 2$, $\;$ common ratio $= r = 3$, $\;$ $n^{th}$ term $= t_n = 4374$
$n^{th}$ term of a G.P $= t_n = ar^{n-1}$
$\therefore \;$ We have, $\;$ $4374 = 2 \times 3^{n-1}$
i.e. $\;$ $3^{n - 1} = \dfrac{4374}{2} = 2187 = 3^7$
$\implies$ $n - 1 = 7$ $\implies$ $n = 8$
i.e. $\;$ Number of terms in the given G.P $= n = 8$
Now, sum to n terms of a G.P $= S_n = \dfrac{a \left(r^n - 1\right)}{r - 1}$, $\;$ $r > 1$
$\therefore \;$ Sum of terms of given G.P is
$S_8 = \dfrac{2 \left(3^{8}-1\right)}{3-1} = \dfrac{2 \times \left(3^8 - 1\right)}{2} = 6560$
$\therefore \;$ $2 + 6 + 18 + 54 + \cdots + 4374 = 6560$