Finding the sum of finite geometric series I'm doing the following summation $\sum_{l=k}^{n}2^l$
$\sum_{l=k}^{n}2^l = 2^k + 2^{k+1} + 2^{k+2} + \ldots+ 2^{n-1} +  2^{n}$
$S_n=a_1\dfrac{1-r^n}{1-r} \therefore S_n=2^k\dfrac{1-(2)^n}{1-2} = 2^{k+n}-2^k$
But my final result seems to be incorrect compared to the one obtained within the calculator
Am i doing something wrong when using the formula above?


The following summation is
$$\sum_{l=k}^{n}2^l = 2^k + 2^{k+1} + 2^{k+2} + \ldots+ 2^{n-1} +  2^{n}$$
can be written as
$$\sum_{l=0}^{n-k}2^{l+k} = 2^k + 2^{k+1} + 2^{k+2} + \ldots+ 2^{n-1} +  2^{n}$$
by shifting the index $k$-times, which is a series with $n−k+1$ terms
$S_{n-k+1}=a_1\dfrac{1-r^{n-k+1}}{1-r} \therefore S_{n-k+1}=2^k\dfrac{1-(2)^{n-k+1}}{1-2} = 2^{n+1}-2^k$
This result seems to be correct compared to the one obtained within the calculator.
