Sequences Nth Term Calculator

Common Sequence Formulas:

\[ \text{Arithmetic: } a_n = a_1 + (n-1)d \] \[ \text{Geometric: } a_n = a_1 \times r^{(n-1)} \] \[ \text{Quadratic: } a_n = an^2 + bn + c \]

Sequence Type:

First Term (a₁):

Common Difference (d):

Quadratic Coefficient (a):

Term Number (n):

Nth Term (aₙ):

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1. What is the Sequences Nth Term Calculator?

This calculator helps you find the nth term of various mathematical sequences including arithmetic, geometric, and quadratic sequences. It's useful for students, teachers, and professionals working with number patterns.

2. How Does the Calculator Work?

The calculator uses standard sequence formulas:

\[ \text{Arithmetic: } a_n = a_1 + (n-1)d \] \[ \text{Geometric: } a_n = a_1 \times r^{(n-1)} \] \[ \text{Quadratic: } a_n = an^2 + bn + c \]

Where:

\( a_n \) — The nth term of the sequence
\( a_1 \) — The first term of the sequence
\( d \) — Common difference (arithmetic)
\( r \) — Common ratio (geometric)
\( a, b, c \) — Coefficients (quadratic)
\( n \) — Term number

3. Types of Sequences

Arithmetic Sequence: Each term increases by a constant difference (e.g., 2, 5, 8, 11... where d=3).
Geometric Sequence: Each term is multiplied by a constant ratio (e.g., 3, 6, 12, 24... where r=2).
Quadratic Sequence: The second differences are constant (e.g., 1, 4, 9, 16... which follows n²).

4. Using the Calculator

Steps: Select sequence type, enter the first term, common difference/ratio, term number, and any additional coefficients needed for quadratic sequences.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between arithmetic and geometric sequences?
A: Arithmetic adds a constant difference each time, while geometric multiplies by a constant ratio.

Q2: How do I identify a quadratic sequence?
A: If the second differences between terms are constant, it's quadratic.

Q3: Can this calculator handle recursive sequences?
A: No, this calculator only handles explicit formulas where the nth term can be calculated directly.

Q4: What if my sequence doesn't fit these types?
A: More complex sequences may require specialized formulas or recursive calculations.

Q5: How accurate are the calculations?
A: Results are accurate to 4 decimal places for precise calculations.