Geometric Progression
How many terms of the G.P $\;$ $1+3+3^2+ \cdots$ $\;$ must be taken to make the sum $3280$?
Let the required number of terms of G.P $= n$
Sum of $n$ terms of G.P $= S_n = \dfrac{a \left(r^n - 1\right)}{r - 1}$ $\;$ when $\;$ $r > 1$
where $\;$ $a = $ first term of G.P and $\;$ $r =$ is its common ratio
Here, $\;$ $a = 1$ $\;$ and $r = 3$
Given: Sum of $n$ terms of G.P $= 3280$
i.e. $\;$ $\dfrac{1 \left(3^n - 1\right)}{3 - 1} = 3280$
i.e. $\;$ $\dfrac{3^n - 1}{2} = 3280$
i.e. $\;$ $3^n - 1 = 6560$
i.e. $\;$ $3^n = 6561 = 3^8$ $\implies$ $n = 8$
i.e. $\;$ Sum of first $8$ terms of the given G.P $= 3280$