Geometric Sequence

1, 3, 9, 27, 81, ... is a geometric sequence: every number in the sequence is obtained by multiplying the previous number by 3.

Some examples of geometric sequences include:

• 3, 6, 12, 24, 48... This sequence has a common ratio of 2 since each term is obtained by multiplying the previous one by 2.
• 5, 20, 80, 320, 1280... This sequence has a common ratio of 4 since each term is obtained by multiplying the previous one by 4.
• 32, 16, 8, 4, 2, 1, 0.5... This sequence has a common ratio of 0.5 since each term is obtained by multiplying the previous one by 0.5.

Find the common ratio for the geometric sequence 6, 18, 54, 162, 486...

Solution:

To find the common ratio of this geometric sequence, take the second term and divide it by the first term of the sequence. To check your result, divide the third term by the second; the fourth by the third; and so on.

$18÷6=3$

$54÷18=3$

$162÷54=3$

$486÷162=3$

Therefore the common ratio for this geometric sequence is 3.

Find the next three terms of the geometric sequence 8, 40, 200, 1000...

Solution:

First, you need to identify the common ratio:

$40÷8=5$

To make sure that the common ratio is 5, check the following terms:

$200÷40=5$

$1000÷200=5$

Now you know that the common ratio is 5, you can use that to find the next terms of the sequence. Just multiply the last term by the common ratio and repeat that to find the next three terms:

$1000\times 5=5000$

$5000\times 5=25000$

$25000\times 5=125000$

Therefore, the next three terms of the sequence are 5000, 25000, 125000

Find the first five terms of the geometric sequence where the first term is 13 and the common ratio is 2.

Solution:

To find each term you can start by multiplying the first term by the term to term rule:

$13\times 2=26$

Now you can continue to multiply the term to term rule by the previous term:

$26\times 2=52$

$52\times 2=104$

$104\times 2=208$

Therefore the first five terms of the sequence are 13, 26, 52, 104, and 208.

Identify the common ratio in the following geometric sequence: 11, 33, 99, 297...

Solution:

To find the common ratio divide the second term of the sequence by the first term of the sequence and so on:

$33÷11=3$

$99÷33=3$

$297÷99=3$

Therefore, the term to term rule of this sequence is 3.

Find the next 3 terms of the geometric sequence 9, 18, 36, 72, 144…

Solution:

First, identify the common ratio:

$18÷9=2$

$36÷18=2$

$72÷36=2$

$144÷72=2$

Now find the next terms by multiplying the common ratio by the previous term;

$144\times 2=288$

$288\times 2=576$

$576\times 2=1152$

Therefore, the next three terms of the sequence are; 288, 576, and 1152.

Find the first five terms of the geometric sequence where 5 is the first term and the common ratio is 4.

Solution:

To do this you need to multiply each term by 4 to find the next:

$5\times 4=20$

$20\times 4=80$

$80\times 4=320$

$320\times 4=1280$

Therefore the first five terms of the sequence are; 5, 20, 80, 320, and 1280.

Identify the term to term rule in the following geometric sequence; 100, 80, 64, 51.2

Solution:

To find the common ratio divide one term by the previous term in the sequence on so on:

$80÷100=0.8or\frac{4}{5}$

$64÷80=0.8or\frac{4}{5}$

$51.2÷64=0.8or\frac{4}{5}$

Therefore, the term to term rule of this sequence is 0.8 or $\frac{4}{5}$.