Geometric Sequence Calculator Form

Geometric Sequence:

N/A

Steps to Use This Geometric Sequence Calculator

  1. Enter the first term (a) of the geometric sequence in the input field labeled "First Term."
  2. Input the common ratio (r) in the field labeled "Common Ratio." The common ratio is the number by which each term is multiplied to get the next term in the sequence.
  3. Enter the number of terms (n) you want to calculate in the field labeled "Number of Terms."
  4. Click the "Calculate Sequence" button to generate the geometric sequence based on the values you provided.
  5. The result will display the sequence in the output section, showing all the terms separated by commas.
  6. If any input is invalid or missing, an error message will appear, prompting you to enter valid numeric values.

What is a Geometric Sequence?

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general form of a geometric sequence can be written as:

\[ a, ar, ar^2, ar^3, \dots \]

Here, \( a \) is the first term of the sequence, and \( r \) is the common ratio. For example, in the sequence 2, 6, 18, 54, each term is multiplied by 3, making the common ratio \( r = 3 \).

Geometric sequences are commonly used in mathematical applications like calculating compound interest, population growth, and even in computer science algorithms.

How to Find a Geometric Sequence on Paper

Follow these steps to find Geometric Sequence for a given first term and common ratio on paper.

  1. Write down the first term (\( a \)) of the sequence. This is the starting point of the geometric sequence.
  2. Identify the common ratio (\( r \)). This is the number you will multiply each term by to get the next term in the sequence.
  3. Multiply the first term by the common ratio to get the second term. Continue multiplying each new term by the common ratio to find the subsequent terms.
  4. Continue this process for as many terms as needed. For example, if the first term is 5 and the common ratio is 2, the sequence will be: 5, 10, 20, 40, 80, and so on.
  5. If you want to find a specific term in the sequence without listing all previous terms, use the formula: \[ a_n = a \times r^{(n-1)} \] where \( a_n \) is the nth term, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term's position in the sequence.

Examples

Example 1: Geometric Sequence with a Common Ratio Greater Than 1

Let’s calculate the first 5 terms of a geometric sequence where the first term \( a = 2 \), the common ratio \( r = 3 \), and the number of terms \( n = 5 \).

Step 1: Write down the formula

The formula for finding the nth term of a geometric sequence is: \[ a_n = a \times r^{(n-1)} \] where: - \( a \) is the first term, - \( r \) is the common ratio, - \( n \) is the term number.

Step 2: Calculate each term

Now, we calculate each term of the sequence using the formula.

- First term: \( a_1 = 2 \times 3^{(1-1)} = 2 \times 1 = 2 \)

- Second term: \( a_2 = 2 \times 3^{(2-1)} = 2 \times 3 = 6 \)

- Third term: \( a_3 = 2 \times 3^{(3-1)} = 2 \times 9 = 18 \)

- Fourth term: \( a_4 = 2 \times 3^{(4-1)} = 2 \times 27 = 54 \)

- Fifth term: \( a_5 = 2 \times 3^{(5-1)} = 2 \times 81 = 162 \)

Step 3: Write the sequence

Therefore, the first 5 terms of the geometric sequence are: 2, 6, 18, 54, 162.

Example 2: Geometric Sequence with a Common Ratio Less Than 1

Let’s calculate the first 5 terms of a geometric sequence where the first term \( a = 100 \), the common ratio \( r = 0.5 \), and the number of terms \( n = 5 \).

Step 1: Write down the formula

We will use the same formula to find the nth term: \[ a_n = a \times r^{(n-1)} \]

Step 2: Calculate each term

Now, we calculate each term of the sequence using the formula.

- First term: \( a_1 = 100 \times 0.5^{(1-1)} = 100 \times 1 = 100 \)

- Second term: \( a_2 = 100 \times 0.5^{(2-1)} = 100 \times 0.5 = 50 \)

- Third term: \( a_3 = 100 \times 0.5^{(3-1)} = 100 \times 0.25 = 25 \)

- Fourth term: \( a_4 = 100 \times 0.5^{(4-1)} = 100 \times 0.125 = 12.5 \)

- Fifth term: \( a_5 = 100 \times 0.5^{(5-1)} = 100 \times 0.0625 = 6.25 \)

Step 3: Write the sequence

Therefore, the first 5 terms of the geometric sequence are: 100, 50, 25, 12.5, 6.25.