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Solution - Geometric Sequences

The common ratio is:

r = 7.619047619047619

r=7.619047619047619

The sum of this series is:

s = 181

s=181

The general form of this series is:

a_{n} = 21 \cdot {7.619047619047619}^{n - 1}

a_n=21*7.619047619047619^(n-1)

The nth term of this series is:

21, 160, 1219.047619047619, 9287.981859410429, 70765.57607169851, 539166.2938796077, 4107933.667654153, 31298542.229745924, 238465083.65520704, 1816876827.8491964

21,160,1219.047619047619,9287.981859410429,70765.57607169851,539166.2938796077,4107933.667654153,31298542.229745924,238465083.65520704,1816876827.8491964

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:

$a_{2} a_{1} = \frac{160}{21} = 7.619047619047619$

The common ratio ( $r$ ) of the sequence is constant and equals the quotient of two consecutive terms.
$r = 7.619047619047619$

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: $a = 21$ , the common ratio: $r = 7.619047619047619$ , and the number of elements $n = 2$ into the geometric series sum formula:

$s_{2} = 21 * ((1 - {7.619047619047619}^{2}) / (1 - 7.619047619047619))$

$s_{2} = 21 * ((1 - 58.04988662131519) / (1 - 7.619047619047619))$

$s_{2} = 21 * (- 57.04988662131519 / (1 - 7.619047619047619))$

$s_{2} = 21 * (- 57.04988662131519 / - 6.619047619047619)$

$s_{2} = 21 \cdot 8.619047619047619$

$s_{2} = 181$

3. Find the general form

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

To find the general form of the series, plug the first term: $a = 21$ and the common ratio: $r = 7.619047619047619$ into the formula for geometric series:

$a_{n} = 21 \cdot {7.619047619047619}^{n - 1}$

4. Find the nth term

Use the general form to find the nth term

$a_{1} = 21$

$a_{2} = a_{1} \cdot r^{n - 1} = 21 \cdot {7.619047619047619}^{2 - 1} = 21 \cdot {7.619047619047619}^{1} = 21 \cdot 7.619047619047619 = 160$

$a_{3} = a_{1} \cdot r^{n - 1} = 21 \cdot {7.619047619047619}^{3 - 1} = 21 \cdot {7.619047619047619}^{2} = 21 \cdot 58.04988662131519 = 1219.047619047619$

$a_{4} = a_{1} \cdot r^{n - 1} = 21 \cdot {7.619047619047619}^{4 - 1} = 21 \cdot {7.619047619047619}^{3} = 21 \cdot 442.28485044811566 = 9287.981859410429$

$a_{5} = a_{1} \cdot r^{n - 1} = 21 \cdot {7.619047619047619}^{5 - 1} = 21 \cdot {7.619047619047619}^{4} = 21 \cdot 3369.789336747548 = 70765.57607169851$

$a_{6} = a_{1} \cdot r^{n - 1} = 21 \cdot {7.619047619047619}^{6 - 1} = 21 \cdot {7.619047619047619}^{5} = 21 \cdot 25674.58542283846 = 539166.2938796077$

$a_{7} = a_{1} \cdot r^{n - 1} = 21 \cdot {7.619047619047619}^{7 - 1} = 21 \cdot {7.619047619047619}^{6} = 21 \cdot 195615.88893591205 = 4107933.667654153$

$a_{8} = a_{1} \cdot r^{n - 1} = 21 \cdot {7.619047619047619}^{8 - 1} = 21 \cdot {7.619047619047619}^{7} = 21 \cdot 1490406.772845044 = 31298542.229745924$

$a_{9} = a_{1} \cdot r^{n - 1} = 21 \cdot {7.619047619047619}^{9 - 1} = 21 \cdot {7.619047619047619}^{8} = 21 \cdot 11355480.174057478 = 238465083.65520704$

$a_{10} = a_{1} \cdot r^{n - 1} = 21 \cdot {7.619047619047619}^{10 - 1} = 21 \cdot {7.619047619047619}^{9} = 21 \cdot 86517944.18329507 = 1816876827.8491964$

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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