Arithmetic Sequence nth Term
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.
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Core idea
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.
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.
Remember it
Memory Aid
Phrase: Any Answer = Always + (Number - 1) Difference
Visual Analogy: Think of a staircase: 'a' is the height of the first step. To reach step 'n', you climb 'n-1' more steps, each with a rise of 'd'.
Exam Tip: Common trap: forgetting the 'minus one'. To find the 20th term, you add the difference 19 times, not 20. Ensure 'd' is negative if values decrease.
Why it makes sense
Intuition
A sequence of discrete points that, if connected, would form a straight line, where the common difference 'd' represents the constant vertical (or horizontal) step size between consecutive points.
Symbols
Variables
a = First Term, d = Common Difference, n = Term Number, a_n = nth Term
Walkthrough
Derivation
Formula for the nth Term of an Arithmetic Sequence
An arithmetic sequence increases or decreases by the same constant difference each term.
- The common difference d is constant.
- n is a positive integer.
Write out the first few terms:
Each term is formed by adding the common difference d.
Spot the pattern:
The nth term equals the first term plus (n − 1) lots of d, since you add d a total of (n − 1) times to reach the nth term.
Note: a₁ = a = first term; d = common difference.
Result
Source: AQA GCSE Maths — Sequences
Where it shows up
Real-World Context
Predict fixed-step savings over weeks.
Avoid these traps
Common Mistakes
- Using n*d instead of (n-1)d.
- Using the wrong first term.
Study smarter
Tips
- Always verify that the difference between the first three terms is identical.
- Remember that n must always be a positive integer representing the term's position.
- Distinguish carefully between the value of the term (an) and the position of the term (n).
Common questions
Frequently Asked Questions
An arithmetic sequence increases or decreases by the same constant difference each term.
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.
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.
Using n*d instead of (n-1)d. Using the wrong first term.
Predict fixed-step savings over weeks.
Always verify that the difference between the first three terms is identical. Remember that n must always be a positive integer representing the term's position. Distinguish carefully between the value of the term (an) and the position of the term (n).