Find the Sum of the Series 1+1/4+1/16+1/64+1/256+1/1024+1/4096

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by ${\displaystyle \frac{1}{4}}$ gives the next term. In other words, ${\displaystyle a_n=a_1{r}^{n-1}}$.

This is the form of a geometric sequence.

Substitute in the values of ${\displaystyle a_1=1}$ and ${\displaystyle r=\frac{1}{4}}$.

Multiply ${\displaystyle {\left(\frac{1}{4}\right)}^{n-1}}$ by ${\displaystyle 1}$.

Apply the product rule to ${\displaystyle \frac{1}{4}}$.

One to any power is one.

This is the formula to find the sum of the first ${\displaystyle n}$ terms of the geometric sequence. To evaluate it, find the values of ${\displaystyle r}$ and ${\displaystyle a_1}$.

Replace the variables with the known values to find ${\displaystyle S_7}$.

Multiply ${\displaystyle \frac{{\left(\frac{1}{4}\right)}^7-1}{\frac{1}{4}-1}}$ by ${\displaystyle 1}$.

Apply the distributive property.

Multiply the numerator by the reciprocal of the denominator.

Multiply ${\displaystyle \frac{-5461}{4096}}$ by ${\displaystyle \frac{1}{-1}}$.

Multiply ${\displaystyle 4096}$ by ${\displaystyle -1}$.