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Penyelesaian - Geometric Sequences

Hasil Akhir Langkah-langkah Terma dan topik

The common ratio is:

r = 10

r=10

The sum of this series is:

s = - 2222

s=-2222

The general form of this series is:

a_{n} = - 2 \cdot 10^{n - 1}

a_n=-2*10^(n-1)

The nth term of this series is:

- 2, - 20, - 200, - 2000, - 20000, - 200000, - 2000000, - 20000000, - 200000000, - 2000000000

-2,-20,-200,-2000,-20000,-200000,-2000000,-20000000,-200000000,-2000000000

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Penjelasan langkah demi langkah

1. Find the common ratio

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

$a_{2} a_{1} = - \frac{20}{-} 2 = 10$

$a_{3} a_{2} = - \frac{200}{-} 20 = 10$

$a_{4} a_{3} = - \frac{2000}{-} 200 = 10$

The common ratio ( $r$ ) of the sequence is constant and equals the quotient of two consecutive terms.
$r = 10$

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: $a = - 2$ , the common ratio: $r = 10$ , and the number of elements $n = 4$ into the geometric series sum formula:

$s_{4} = - 2 * ((1 - 10^{4}) / (1 - 10))$

$s_{4} = - 2 * ((1 - 10000) / (1 - 10))$

$s_{4} = - 2 * (- 9999 / (1 - 10))$

$s_{4} = - 2 * (- 9999 / - 9)$

$s_{4} = - 2 \cdot 1111$

$s_{4} = - 2222$

3. Find the general form

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

To find the general form of the series, plug the first term: $a = - 2$ and the common ratio: $r = 10$ into the formula for geometric series:

$a_{n} = - 2 \cdot 10^{n - 1}$

4. Find the nth term

Use the general form to find the nth term

$a_{1} = - 2$

$- 2 \cdot 10^{2 - 1} = - 2 \cdot 10^{1} = - 2 \cdot 10 = - 20$

$- 2 \cdot 10^{3 - 1} = - 2 \cdot 10^{2} = - 2 \cdot 100 = - 200$

$- 2 \cdot 10^{4 - 1} = - 2 \cdot 10^{3} = - 2 \cdot 1000 = - 2000$

$- 2 \cdot 10^{5 - 1} = - 2 \cdot 10^{4} = - 2 \cdot 10000 = - 20000$

$- 2 \cdot 10^{6 - 1} = - 2 \cdot 10^{5} = - 2 \cdot 100000 = - 200000$

$- 2 \cdot 10^{7 - 1} = - 2 \cdot 10^{6} = - 2 \cdot 1000000 = - 2000000$

$- 2 \cdot 10^{8 - 1} = - 2 \cdot 10^{7} = - 2 \cdot 10000000 = - 20000000$

$- 2 \cdot 10^{9 - 1} = - 2 \cdot 10^{8} = - 2 \cdot 100000000 = - 200000000$

$- 2 \cdot 10^{10 - 1} = - 2 \cdot 10^{9} = - 2 \cdot 1000000000 = - 2000000000$

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Learn more with Tiger

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.

Terma dan topik

Geometric Sequences