Algebra Examples

2

$2$ ,

4

$4$ ,

8

$8$ ,

16

$16$ ,

32

$32$

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

$2$ gives the next term. In other words,

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

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

Geometric Sequence:

r = 2

$r = 2$

Step 2

This is the form of a geometric sequence.

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

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

Step 3

Substitute in the values of

a_{1} = 2

$a_{1} = 2$ and

r = 2

$r = 2$ .

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

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

Step 4

Multiply

2

$2$ by

2^{n - 1}

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

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Step 4.1

Multiply

2

$2$ by

2^{n - 1}

$2^{n - 1}$ .

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Step 4.1.1

Raise

2

$2$ to the power of

1

$1$ .

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

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

Step 4.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

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

Step 4.2

Combine the opposite terms in

1 + n - 1

$1 + n - 1$ .

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Step 4.2.1

Subtract

1

$1$ from

1

$1$ .

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

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

Step 4.2.2

Add

n

$n$ and

0

$0$ .

a_{n} = 2^{n}

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

a_{n} = 2^{n}

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

a_{n} = 2^{n}

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