Calculus Examples

$5$ , $10$ , $20$ , $40$

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 $a_{1} = 5$ and $r = 2$ .

$a_{n} = 5 \cdot 2^{n - 1}$

Step 4

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 5

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

$S_{4} = 5 \cdot \frac{{(2)}^{4} - 1}{2 - 1}$

Step 6

Simplify the numerator.

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Step 6.1

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

$S_{4} = 5 \cdot \frac{16 - 1}{2 - 1}$

Step 6.2

Subtract $1$ from $16$ .

$S_{4} = 5 \cdot \frac{15}{2 - 1}$

Step 7

Simplify the expression.

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Step 7.1

Subtract $1$ from $2$ .

$S_{4} = 5 \cdot \frac{15}{1}$

Step 7.2

Divide $15$ by $1$ .

$S_{4} = 5 \cdot 15$

Step 7.3

Multiply $5$ by $15$ .

$S_{4} = 75$

Step 8

Convert the fraction to a decimal.

$S_{4} = 75$