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Solution - Geometric Sequences

The common ratio is:

r = 0.2222222222222222

r=0.2222222222222222

The sum of this series is:

s = 109

s=109

The general form of this series is:

a_{n} = 90 \cdot {0.2222222222222222}^{n - 1}

a_n=90*0.2222222222222222^(n-1)

The nth term of this series is:

90, 20, 4.444444444444445, 0.987654320987654, 0.21947873799725648, 0.04877305288827921, 0.01083845619739538, 0.0024085458216434176, 0.0005352324048096483, 0.00011894053440214406

90,20,4.444444444444445,0.987654320987654,0.21947873799725648,0.04877305288827921,0.01083845619739538,0.0024085458216434176,0.0005352324048096483,0.00011894053440214406

See steps

Step-by-step explanation

1. Find the common ratio

Find the common ratio by dividing any term in the sequence by the term that comes before it:

$a_{2} a_{1} = \frac{20}{90} = 0.2222222222222222$

The common ratio ( $r$ ) of the sequence is constant and equals the quotient of two consecutive terms.
$r = 0.2222222222222222$

2. Find the sum

5 additional steps

$s_{n} = a * ((1 - r^{n}) / (1 - r))$

To find the sum of the series, plug the first term: $a = 90$ , the common ratio: $r = 0.2222222222222222$ , and the number of elements $n = 2$ into the geometric series sum formula:

$s_{2} = 90 * ((1 - {0.2222222222222222}^{2}) / (1 - 0.2222222222222222))$

$s_{2} = 90 * ((1 - 0.04938271604938271) / (1 - 0.2222222222222222))$

$s_{2} = 90 * (0.9506172839506173 / (1 - 0.2222222222222222))$

$s_{2} = 90 * (0.9506172839506173 / 0.7777777777777778)$

$s_{2} = 90 \cdot 1.222222222222222$

$s_{2} = 109.99999999999999$

3. Find the general form

$a_{n} = a \cdot r^{n - 1}$

To find the general form of the series, plug the first term: $a = 90$ and the common ratio: $r = 0.2222222222222222$ into the formula for geometric series:

$a_{n} = 90 \cdot {0.2222222222222222}^{n - 1}$

4. Find the nth term

Use the general form to find the nth term

$a_{1} = 90$

$a_{2} = a_{1} \cdot r^{n - 1} = 90 \cdot {0.2222222222222222}^{2 - 1} = 90 \cdot {0.2222222222222222}^{1} = 90 \cdot 0.2222222222222222 = 20$

$a_{3} = a_{1} \cdot r^{n - 1} = 90 \cdot {0.2222222222222222}^{3 - 1} = 90 \cdot {0.2222222222222222}^{2} = 90 \cdot 0.04938271604938271 = 4.444444444444445$

$a_{4} = a_{1} \cdot r^{n - 1} = 90 \cdot {0.2222222222222222}^{4 - 1} = 90 \cdot {0.2222222222222222}^{3} = 90 \cdot 0.010973936899862823 = 0.987654320987654$

$a_{5} = a_{1} \cdot r^{n - 1} = 90 \cdot {0.2222222222222222}^{5 - 1} = 90 \cdot {0.2222222222222222}^{4} = 90 \cdot 0.002438652644413961 = 0.21947873799725648$

$a_{6} = a_{1} \cdot r^{n - 1} = 90 \cdot {0.2222222222222222}^{6 - 1} = 90 \cdot {0.2222222222222222}^{5} = 90 \cdot 0.000541922809869769 = 0.04877305288827921$

$a_{7} = a_{1} \cdot r^{n - 1} = 90 \cdot {0.2222222222222222}^{7 - 1} = 90 \cdot {0.2222222222222222}^{6} = 90 \cdot 0.00012042729108217089 = 0.01083845619739538$

$a_{8} = a_{1} \cdot r^{n - 1} = 90 \cdot {0.2222222222222222}^{8 - 1} = 90 \cdot {0.2222222222222222}^{7} = 90 \cdot 2.6761620240482418 E - 05 = 0.0024085458216434176$

$a_{9} = a_{1} \cdot r^{n - 1} = 90 \cdot {0.2222222222222222}^{9 - 1} = 90 \cdot {0.2222222222222222}^{8} = 90 \cdot 5.947026720107203 E - 06 = 0.0005352324048096483$

$a_{10} = a_{1} \cdot r^{n - 1} = 90 \cdot {0.2222222222222222}^{10 - 1} = 90 \cdot {0.2222222222222222}^{9} = 90 \cdot 1.3215614933571563 E - 06 = 0.00011894053440214406$

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Why learn this

Geometric sequences are commonly used to explain concepts in mathematics, physics, engineering, biology, economics, computer science, finance, and more, making them a very useful tool to have in our toolkits. One of the most common applications of geometric sequences, for example, is calculating earned or unpaid compound interest, an activity most commonly associated with finance that could mean earning or losing a lot of money! Other applications include, but are certainly not limited to, calculating probability, measuring radioactivity over time, and designing buildings.

Terms and topics

Geometric Sequences