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Penyelesaian - Geometric Sequences

Hasil Akhir Langkah-langkah Terma dan topik

The common ratio is:

r = 4.105263157894737

r=4.105263157894737

The sum of this series is:

s = 7469

s=7469

The general form of this series is:

a_{n} = 1463 \cdot {4.105263157894737}^{n - 1}

a_n=1463*4.105263157894737^(n-1)

The nth term of this series is:

1463, 6006.000000000001, 24656.21052631579, 101220.23268698064, 415535.6920833942, 1705883.36750025, 7003100.140264185, 28749568.996874027, 118024546.4082197, 484521822.0969019

1463,6006.000000000001,24656.21052631579,101220.23268698064,415535.6920833942,1705883.36750025,7003100.140264185,28749568.996874027,118024546.4082197,484521822.0969019

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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:

$a_{2} a_{1} = \frac{6006}{1463} = 4.105263157894737$

The common ratio ( $r$ ) of the sequence is constant and equals the quotient of two consecutive terms.
$r = 4.105263157894737$

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: $a = 1, 463$ , the common ratio: $r = 4.105263157894737$ , and the number of elements $n = 2$ into the geometric series sum formula:

$s_{2} = 1463 * ((1 - {4.105263157894737}^{2}) / (1 - 4.105263157894737))$

$s_{2} = 1463 * ((1 - 16.85318559556787) / (1 - 4.105263157894737))$

$s_{2} = 1463 * (- 15.853185595567869 / (1 - 4.105263157894737))$

$s_{2} = 1463 * (- 15.853185595567869 / - 3.105263157894737)$

$s_{2} = 1463 \cdot 5.105263157894737$

$s_{2} = 7469.000000000001$

3. Find the general form

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

To find the general form of the series, plug the first term: $a = 1, 463$ and the common ratio: $r = 4.105263157894737$ into the formula for geometric series:

$a_{n} = 1463 \cdot {4.105263157894737}^{n - 1}$

4. Find the nth term

Use the general form to find the nth term

$a_{1} = 1463$

$1463 \cdot {4.105263157894737}^{2 - 1} = 1463 \cdot {4.105263157894737}^{1} = 1463 \cdot 4.105263157894737 = 6006.000000000001$

$1463 \cdot {4.105263157894737}^{3 - 1} = 1463 \cdot {4.105263157894737}^{2} = 1463 \cdot 16.85318559556787 = 24656.21052631579$

$1463 \cdot {4.105263157894737}^{4 - 1} = 1463 \cdot {4.105263157894737}^{3} = 1463 \cdot 69.18676191864705 = 101220.23268698064$

$1463 \cdot {4.105263157894737}^{5 - 1} = 1463 \cdot {4.105263157894737}^{4} = 1463 \cdot 284.0298647186563 = 415535.6920833942$

$1463 \cdot {4.105263157894737}^{6 - 1} = 1463 \cdot {4.105263157894737}^{5} = 1463 \cdot 1166.017339371326 = 1705883.36750025$

$1463 \cdot {4.105263157894737}^{7 - 1} = 1463 \cdot {4.105263157894737}^{6} = 1463 \cdot 4786.80802478755 = 7003100.140264185$

$1463 \cdot {4.105263157894737}^{8 - 1} = 1463 \cdot {4.105263157894737}^{7} = 1463 \cdot 19651.106628075206 = 28749568.996874027$

$1463 \cdot {4.105263157894737}^{9 - 1} = 1463 \cdot {4.105263157894737}^{8} = 1463 \cdot 80672.96405209822 = 118024546.4082197$

$1463 \cdot {4.105263157894737}^{10 - 1} = 1463 \cdot {4.105263157894737}^{9} = 1463 \cdot 331183.7471612453 = 484521822.0969019$

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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