Algebra Examples

$- 2$ , $- 8$ , $- 32$ , $- 128$

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: $r = 4$

This is the form of a geometric sequence.

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

Substitute in the values of $a_{1} = - 2$ and $r = 4$ .

$a_{n} = - 2 \cdot 4^{n - 1}$

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

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Factor out negative.

$a_{n} = - (2 \cdot 4^{n - 1})$

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

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

Multiply the exponents in ${(2^{2})}^{n - 1}$ .

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Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

$a_{n} = - (2 \cdot 2^{2 n - 2})$

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

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

Subtract $2$ from $1$ .

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

Substitute in the value of $n$ to find the $n$ th term.

$a_{6} = - 2^{2 (6) - 1}$

Multiply $2$ by $6$ .

$a_{6} = - 2^{12 - 1}$

Subtract $1$ from $12$ .

$a_{6} = - 2^{11}$

Raise $2$ to the power of $11$ .

$a_{6} = - 1 \cdot 2048$

Multiply $- 1$ by $2048$ .

$a_{6} = - 2048$