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Geometric Sequence nth Term Calculator

Find aₙ for a geometric sequence.

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nth Term

Formula first

Overview

The geometric sequence nth term formula determines the value of any specific member in a progression where each subsequent term is derived by multiplying the preceding one by a constant ratio. This exponential relationship allows for efficient calculation of values in a sequence without requiring the manual listing of every intermediate step.

Symbols

Variables

a = First Term, r = Common Ratio, n = Term Number, = nth Term

First Term
Variable
Common Ratio
Variable
Term Number
Variable
nth Term
Variable

Apply it well

When To Use

When to use: Use this formula when a value changes by a constant percentage or fixed multiplier over discrete, equal intervals. It is ideal for calculating future values in population growth models, financial interest scenarios, or physical processes like radioactive decay where the change is proportional to the current amount.

Why it matters: Understanding this formula is essential for modeling real-world phenomena that exhibit exponential growth or decay. It allows economists to project wealth over time, biologists to predict bacterial colony sizes, and engineers to understand signal attenuation in communication systems.

Avoid these traps

Common Mistakes

  • Using addition instead of multiplication.
  • Wrong exponent on r.

One free problem

Practice Problem

Practice Problem 1

A geometric sequence begins with 5 and has a common ratio of 3. Calculate the value of the 6th term.

First Term5
Common Ratio3
Term Number6

Solve for: an

Hint: Identify the exponent first by subtracting 1 from the term position.

Practice Problem 2

In a geometric sequence where the common ratio is 2, the 4th term is found to be 80. Determine the first term of this sequence.

nth Term80
Common Ratio2
Term Number4

Solve for: an

Hint: Divide the value of the nth term by the ratio raised to the (n-1) power.

Practice Problem 3

The first term of a sequence is 4 and the 3rd term is 100. Assuming the common ratio is positive, what is its value?

First Term4
nth Term100
Term Number3

Solve for: an

Hint: Isolate the ratio by dividing the nth term by the first term, then find the square root.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Geometric progression
  2. IUPAC Gold Book
  3. Edexcel GCSE Maths — Sequences