Precalculus Examples

$3 + 12 + 48 + 192 + 768$

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

Geometric Sequence: $r = 4$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = 3$ and $r = 4$ .

$a_{n} = 3 \cdot 4^{n - 1}$

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 $S_{5}$ .

$S_{5} = 3 \cdot \frac{{(4)}^{5} - 1}{4 - 1}$

Step 6

Simplify the numerator.

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Step 6.1

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

$S_{5} = 3 \cdot \frac{1024 - 1}{4 - 1}$

Step 6.2

Subtract $1$ from $1024$ .

$S_{5} = 3 \cdot \frac{1023}{4 - 1}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Subtract $1$ from $4$ .

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

Step 7.2

Cancel the common factor of $3$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

$S_{5} = 1023$

Step 8

Convert the fraction to a decimal.

$S_{5} = 1023$