What is the sum of the first $20$ terms of the geometric sequence $7, 14, 28, 56, 112, 224, \ldots$?

The sequence is geometric with $a_{1} = 7$ and $r = 2$. Finite geometric sum formula (for $r \ne 1$): $S_{n} = \dfrac{a_{1}(1 - r^{n})}{1 - r}$. With $a_{1} = 7$, $r = 2$, $n = 20$: $S_{20} = \dfrac{7(1 - 2^{20})}{1 - 2} = \dfrac{7(1 - 1{,}048{,}576)}{-1} = \dfrac{7(-1{,}048{,}575)}{-1} = 7 \times 1{,}048{,}575 = 7{,}340{,}025$. So $S_{20} = 7(2^{20} - 1) = 7{,}340{,}025$. - Trap B ($7 \cdot 2^{19}$) is just the **20th term** $a_{20}$, not the sum of the first 20 terms. - Trap D ($7 \cdot 2^{20}$) drops the $-1$ in the geometric-sum closed form. - Trap A treats the sum as if the sequence were constant. For $r = 2$, the equivalent simplification is $S_{n} = a_{1}(2^{n} - 1)$, which makes the $-1$ in option C the load-bearing detail.