How to Use the Sequence Calculator

A Sequence Calculator is a digital tool that generates arithmetic or geometric sequences, calculates the n-th term, and finds the sum of the first n terms of the sequence.

What It Does

This calculator saves time by automatically computing values for common mathematical sequences. Instead of manually calculating terms, you can input the values and instantly generate:

• The full list of terms in the sequence
• The n-th term of the sequence
• The sum of the first n terms

It works for:

1. Arithmetic sequences (where each term increases by a fixed difference, e.g., 2, 5, 8, 11… with common difference d = 3).
2. Geometric sequences (where each term multiplies by a fixed ratio, e.g., 3, 6, 12, 24… with common ratio r = 2).

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Steps to Use the Sequence Calculator

1. Choose the type of sequence:
   • Select Arithmetic if the numbers increase by addition/subtraction.
   • Select Geometric if the numbers increase by multiplication/division.
2. Enter the first term (a₁):
   This is the starting number of the sequence.
3. Enter the common difference or ratio (d or r):
   • For arithmetic, this is the number added/subtracted each step.
   • For geometric, this is the factor by which you multiply/divide each step.
4. Enter the number of terms (n):
   This is how many terms you want to calculate.
5. Click “Generate Sequence”:
   The calculator will display:
   • The complete sequence
   • The value of the n-th term
   • The sum of n terms

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Example

Suppose you want the first 10 terms of an arithmetic sequence starting with 2 and a common difference of 3:

• Input: Sequence type = Arithmetic, First Term = 2, Common Difference = 3, Terms = 10
• Output:
  • Sequence: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29
  • 10th Term = 29
  • Sum = 155

For a geometric sequence with a₁ = 3, r = 2, n = 5:

• Output:
  • Sequence: 3, 6, 12, 24, 48
  • 5th Term = 48
  • Sum = 93

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Sequence Calculator FAQ

Q1: What is the difference between arithmetic and geometric sequences?
A: Arithmetic sequences change by addition/subtraction of a fixed number, while geometric sequences change by multiplication/division of a fixed number.

Q2: What formulas does this calculator use?

• Arithmetic nth term: aₙ = a₁ + (n-1)d
• Arithmetic sum: Sₙ = (n/2)(2a₁ + (n-1)d)
• Geometric nth term: aₙ = a₁ × rⁿ⁻¹
• Geometric sum: Sₙ = a₁( rⁿ - 1 ) / (r - 1), if r ≠ 1

Q3: What happens if I set the common ratio to 1 in a geometric sequence?
A: All terms will be the same as the first term, and the sum will be simply the first term × number of terms.

Q4: Can this calculator handle negative values?
A: Yes, you can use negative first terms, differences, or ratios. The sequence will be generated accordingly.

Q5: Is this calculator suitable for students and teachers?
A: Absolutely! It’s a helpful tool for quick verification of problems, homework support, and teaching demonstrations.