Find the Sum of the Series 4 , 16 , 64 , 256 , 1024

$4$ , $16$ , $64$ , $256$ , $1024$

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 $4$ gives the next term. In other words, $a_{n} = a_{1} \cdot r^{n - 1}$ .

Geometric Sequence: $r = 4$

This is the form of a geometric sequence.

$a_{n} = a_{1} r^{n - 1}$

Substitute in the values of $a_{1} = 4$ and $r = 4$ .

$a_{n} = (4) \cdot {(4)}^{n - 1}$

Multiply $4$ by ${(4)}^{n - 1}$ by adding the exponents.

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Multiply $4$ by ${(4)}^{n - 1}$ .

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Raise $4$ to the power of $1$ .

$a_{n} = 4 \cdot {(4)}^{n - 1}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a_{n} = 4^{1 + n - 1}$

Combine the opposite terms in $1 + n - 1$ .

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Subtract $1$ from $1$ .

$a_{n} = 4^{n + 0}$

Add $n$ and $0$ .

$a_{n} = 4^{n}$

This is the formula to find the sum of the first $n$ terms of the geometric sequence. To evaluate it, find the values of $r$ and $a_{1}$ .

$S_{n} = \frac{a_{1} (r^{n} - 1)}{r - 1}$

Replace the variables with the known values to find $S_{5}$ .

$S_{5} = 4 \cdot \frac{{(4)}^{5} - 1}{4 - 1}$

Simplify the numerator.

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Raise $4$ to the power of $5$ .

$S_{5} = 4 \cdot \frac{1024 - 1}{4 - 1}$

Subtract $1$ from $1024$ .

$S_{5} = 4 \cdot \frac{1023}{4 - 1}$

Simplify the expression.

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Subtract $1$ from $4$ .

$S_{5} = 4 \cdot \frac{1023}{3}$

Divide $1023$ by $3$ .

$S_{5} = 4 \cdot 341$

Multiply $4$ by $341$ .

$S_{5} = 1364$

Convert the fraction to a decimal.

$S_{5} = 1364$

Find the Sum of the Series 4 , 16 , 64 , 256 , 1024

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