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Select and enter the values to calculate the geometric progression and related parameters in the sequence.

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The geometric sequence calculator is used to find geometric series step by step along with graphical representation. Using this calculator allows you to find other values of the geometric sequence including:

- First term (a₁)
- Common ratio (r)
- nᵗʰ number of terms
- Sum of the first (n) terms

**Limitation: **The calculator cannot handle infinite geometric sequences and it works only with real numbers. It cannot handle complex numbers as common ratios.

- Select the parameter that you want to calculate
- Enter the values into the given boxes
- Click
**‘**Calculate**’** - Get the results

**“The sequence of numbers where each term (except the first term) is derived by multiplying the previous term with a constant non-zero number (common ratio)”**

To find the preceding term in the given sequence, divide the term by the same common ratio (r).

\(\ a_{n} = {a_{1}\times(r^{n - 1})}\)

**Where:**

- an = nth term of the sequence
- a₁ = first term of the sequence
- r = common ratio between successive terms
- n = number of terms in the series

**Key Elements:**

**nᵗʰ term of series:**It refers to the value of any term in the sequence, where 'n' represents the term's position.**First Term (a₁):**The starting point of the sequence from which all the subsequent terms are derived.**Common Ratio (r):**The constant term that determines the sequence from one term to the other. It is the number that is repeatedly multiplied by each term in a series.

To calculate the geometric sequence, multiply the first term of the sequence by the common ratio raised to the power of position ‘n’ minus one (n-1).

Consider the sequence 3, 6, 12, 24, … Find the 5th term in the sequence and Sum of the first n-terms.

**Given Values:**

- The first term (a₁) = 3
- The common ratio (r) = 2
- n = 5

**Find the 5th term (n = 5) in the sequence:**

\(\ a_n = a_1 * r^{n-1}\)

\(\ a_{5} = (3)*(2)^{5 - 1}\)

\(\ a_{5} = (3)*(2)^{4}\)

\(\ a_{5} = (3)*(16)\)

\(\ a_{5} = 48\)

Therefore, the 5th term in the sequence is 48.

**Find the sum of the first n-terms:**

\(\S_n = a + ar + ar^2 + ar^3... + ar^{n-1}\)

\(\S_5 = 3 + 6 + 12 + 24 + 48\)

\(\ S_{5} = 93\)

So, the sum of the first n-terms equals to 93. To get the answers instantly, you can simply enter the first term, common ratio, and number of terms into the geometric progression calculator.

**Finite Geometric Sequence: **Geometric sequence with a finite number of terms.

\(\ S_{n}=\frac{a_{1}\times(1-r^{n})}{1-r}\)

**Infinite Geometric Sequence: **Geometric sequence with an infinite number of terms.

\(\ S_{\infty} = \frac{a}{1-r}\)

Geometric sequences are commonly used in everyday situations with a key usage in calculating interest. Its other applications include:

- Modeling population growth
- Exponential time algorithms
- Understanding exponential patterns
- Trend forecasting, Investment growth
- Population growth, disease spread
- Signal attenuation, amplification

Here are the steps for finding the sum of finite geometric series:

- Calculate the common ratio raised to the power of n (r^n)
- Take the result from ‘step 1’ and subtract 1 from it
- Divide the result by (1 - r)
- Multiply the conclusion by (a₁) of the sequence

The common ratio is obtained by dividing any term by the preceding term. It determines how the sequence progresses. While you can verify that there is a common ratio by dividing several terms, a geometric sequence calculator makes this calculation and shows that all the terms are constant.

\(\ r = \frac{a(n + 1)}{an}\)

To determine the nth term in a geometric sequence, follow the steps below:

- Find the common ratio ‘r’ to the power of ‘n-1’ (r^(n-1))
- Multiply the resultant answer from step 1 by 1st term (a₁)

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