Precalculus Examples

$2$ , $4$ , $8$

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

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

Step 4

Multiply $2$ by $2^{n - 1}$ by adding the exponents.

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Step 4.1

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

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Step 4.1.1

Raise $2$ to the power of $1$ .

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

Step 4.1.2

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

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

Step 4.2

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

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Step 4.2.1

Subtract $1$ from $1$ .

$a_{n} = 2^{n + 0}$

Step 4.2.2

Add $n$ and $0$ .

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

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 $S_{3}$ .

$S_{3} = 2 \cdot \frac{{(2)}^{3} - 1}{2 - 1}$

Step 7

Simplify the numerator.

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Step 7.1

Raise $2$ to the power of $3$ .

$S_{3} = 2 \cdot \frac{8 - 1}{2 - 1}$

Step 7.2

Subtract $1$ from $8$ .

$S_{3} = 2 \cdot \frac{7}{2 - 1}$

Step 8

Simplify the expression.

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Step 8.1

Subtract $1$ from $2$ .

$S_{3} = 2 \cdot \frac{7}{1}$

Step 8.2

Divide $7$ by $1$ .

$S_{3} = 2 \cdot 7$

Step 8.3

Multiply $2$ by $7$ .

$S_{3} = 14$

Step 9

Convert the fraction to a decimal.

$S_{3} = 14$