Calculus Examples

1

$1$ ,

2

$2$ ,

4

$4$ ,

8

$8$ ,

16

$16$ ,

32

$32$ ,

64

$64$ ,

128

$128$ ,

256

$256$ ,

512

$512$ ,

1024

$1024$ ,

2048

$2048$ ,

4096

$4096$

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} = 1

$a_{1} = 1$ and

r = 2

$r = 2$ .

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

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

Step 4

Multiply

2^{n - 1}

$2^{n - 1}$ by

1

$1$ .

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

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

Step 5

Substitute in the value of

n

$n$ to find the

n

$n$ th term.

a_{14} = 2^{(14) - 1}

$a_{14} = 2^{(14) - 1}$

Step 6

Subtract

1

$1$ from

14

$14$ .

a_{14} = 2^{13}

$a_{14} = 2^{13}$

Step 7

Raise

2

$2$ to the power of

13

$13$ .

a_{14} = 8192

$a_{14} = 8192$

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096

$1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096$

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