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Arithmetic Sequence nth Term Calculator

Find aₙ for an arithmetic sequence.

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nth Term

Formula first

Overview

This formula identifies any specific term within an arithmetic progression where the difference between consecutive terms remains constant. It utilizes the starting value and a linear growth pattern to calculate the value at any discrete position without manual counting.

Symbols

Variables

a = First Term, d = Common Difference, n = Term Number, = nth Term

First Term
Variable
Common Difference
Variable
Term Number
Variable
nth Term
Variable

Apply it well

When To Use

When to use: Use this equation when dealing with patterns that increase or decrease by a fixed amount at each step. It assumes the sequence is linear and discrete, meaning the common difference remains unchanged throughout the set.

Why it matters: It is foundational for financial calculations like simple interest and straight-line depreciation, as well as predicting future states in systems with steady growth. In computer science, it helps determine memory addresses and loop iterations.

Avoid these traps

Common Mistakes

  • Using n*d instead of (n-1)d.
  • Using the wrong first term.

One free problem

Practice Problem

Practice Problem 1

A construction crew builds 5 meters of a wall on the first day and increases their daily output by 3 meters each subsequent day. How many meters will they build on the 12th day?

First Term5
Common Difference3
Term Number12

Solve for: an

Hint: Identify the initial amount (a) and the daily increase (d), then substitute them into the formula for n = 12.

Practice Problem 2

A theater has 20 seats in the first row. Each following row has 4 more seats than the previous one. If a specific row contains 100 seats, which row number is it?

First Term20
Common Difference4
nth Term100

Solve for:

Hint: Plug the known values into the nth term formula and solve the resulting linear equation for n.

Practice Problem 3

An investment account starts with 2580, what is the monthly increase?

First Term1500
nth Term2580
Term Number10

Solve for:

Hint: Subtract the initial balance from the 10th-month balance and divide by the number of steps (n-1).

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Arithmetic progression
  2. Britannica: Arithmetic progression
  3. AQA GCSE Maths — Sequences