Find the Sum of the First 6 Terms 4 , 8 , 16 , 32

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

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

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

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

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

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

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 $S_{6}$ .

$S_{6} = 4 \cdot \frac{{(2)}^{6} - 1}{2 - 1}$

Step 7

Simplify the numerator.

Raise $2$ to the power of $6$ .

$S_{6} = 4 \cdot \frac{64 - 1}{2 - 1}$

Subtract $1$ from $64$ .

$S_{6} = 4 \cdot \frac{63}{2 - 1}$

Step 8

Simplify the expression.

Subtract $1$ from $2$ .

$S_{6} = 4 \cdot \frac{63}{1}$

Divide $63$ by $1$ .

$S_{6} = 4 \cdot 63$

Multiply $4$ by $63$ .

$S_{6} = 252$

Step 9

Convert the fraction to a decimal.

$S_{6} = 252$

Find the Sum of the First 6 Terms 4 , 8 , 16 , 32

Related Questions

Support LCMGCF.com by disabling your adblocker.