Calculus Examples

$4$ , $8$ , $16$ , $32$

Step 1

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 $2$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = 2$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = 4$ and $r = 2$ .

$a_{n} = 4 \cdot 2^{n - 1}$

Step 4

Multiply $4 \cdot 2^{n - 1}$ .

Tap for more steps...

Step 4.1

Rewrite $4$ as $2^{2}$ .

$a_{n} = 2^{2} \cdot 2^{n - 1}$

Step 4.2

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

$a_{n} = 2^{2 + n - 1}$

Step 4.3

Subtract $1$ from $2$ .

$a_{n} = 2^{n + 1}$

Step 5

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 $a_{1}$ .

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

Step 6

Replace the variables with the known values to find $S_{4}$ .

$S_{4} = 4 \cdot \frac{{(2)}^{4} - 1}{2 - 1}$

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

Raise $2$ to the power of $4$ .

$S_{4} = 4 \cdot \frac{16 - 1}{2 - 1}$

Step 7.2

Subtract $1$ from $16$ .

$S_{4} = 4 \cdot \frac{15}{2 - 1}$

Step 8

Simplify the expression.

Tap for more steps...

Step 8.1

Subtract $1$ from $2$ .

$S_{4} = 4 \cdot \frac{15}{1}$

Step 8.2

Divide $15$ by $1$ .

$S_{4} = 4 \cdot 15$

Step 8.3

Multiply $4$ by $15$ .

$S_{4} = 60$

Step 9

Convert the fraction to a decimal.

$S_{4} = 60$