Identify the Progression 4 , 16 , 64 , 256

$4$ , $16$ , $64$ , $256$

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 $a_{1} = 4$ and $r = 4$ .

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

Multiply $4$ by $4^{n - 1}$ by adding the exponents.

Multiply $4$ by $4^{n - 1}$ .

Raise $4$ to the power of $1$ .

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

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

$a_{n} = 4^{1 + n - 1}$

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

Subtract $1$ from $1$ .

$a_{n} = 4^{n + 0}$

Add $n$ and $0$ .

$a_{n} = 4^{n}$

Identify the Progression 4 , 16 , 64 , 256

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