Calculus Examples

$1$ , $2$ , $4$ , $8$ , $16$

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 $a_{1} = 1$ and $r = 2$ .

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

Step 4

Multiply $2^{n - 1}$ by $1$ .

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

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 $S_{7}$ .

$S_{7} = 1 \cdot \frac{{(2)}^{7} - 1}{2 - 1}$

Step 7

Multiply $\frac{{(2)}^{7} - 1}{2 - 1}$ by $1$ .

$S_{7} = \frac{{(2)}^{7} - 1}{2 - 1}$

Step 8

Simplify the numerator.

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Step 8.1

Raise $2$ to the power of $7$ .

$S_{7} = \frac{128 - 1}{2 - 1}$

Step 8.2

Subtract $1$ from $128$ .

$S_{7} = \frac{127}{2 - 1}$

Step 9

Simplify the expression.

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Step 9.1

Subtract $1$ from $2$ .

$S_{7} = \frac{127}{1}$

Step 9.2

Divide $127$ by $1$ .

$S_{7} = 127$

Step 10

Convert the fraction to a decimal.

$S_{7} = 127$

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