Arithmetic Sequence (n-th Term)

Statement

An arithmetic sequence is a sequence of numbers where each term is obtained by adding the same constant difference to the previous term. The general formula for the \(n\)-th term is:

\[ u_n = a + (n-1)d \]

Here, \(a\) is the first term, \(d\) is the common difference, and \(n\) is the position of the term in the sequence.

Why it’s true (short reason)

• Each step in the sequence increases (or decreases) by the common difference \(d\).
• The second term is \(a+d\), the third is \(a+2d\), the fourth is \(a+3d\), and so on.
• By the time you reach the \(n\)-th term, you have added the difference \((n-1)\) times.
• This gives the formula \(u_n = a + (n-1)d\).

Recipe (how to use it)

1. Identify the first term \(a\).
2. Find the common difference \(d\) (subtract one term from the next).
3. Decide which term \(u_n\) you want (e.g. 10th term means \(n=10\)).
4. Substitute into the formula: \(u_n = a + (n-1)d\).
5. Simplify to find the required term.

Spotting it

Use this formula when:

• The sequence has a constant difference between consecutive terms.
• You are asked for a specific term far into the sequence.
• You want to express the sequence in algebraic form.

Common pairings

• The arithmetic series sum formula \(S_n = \tfrac{n}{2}(2a+(n-1)d)\).
• Linear functions, since arithmetic sequences can be expressed as straight-line rules.
• Word problems involving patterns, steps, or repeated increases/decreases.

Mini examples

1. Sequence: 3, 7, 11, 15, … First term \(a=3\), common difference \(d=4\). 10th term: \(u_{10} = 3 + (10-1)\times 4 = 39\).
2. Sequence: 20, 18, 16, 14, … First term \(a=20\), common difference \(d=-2\). 12th term: \(u_{12} = 20 + (12-1)\times(-2) = -2\).

Pitfalls

• Forgetting that the difference may be negative.
• Using \(n\) instead of \(n-1\) when multiplying by the difference.
• Confusing arithmetic sequences with geometric ones (which multiply instead of add).
• Not simplifying properly after substitution.

Exam strategy

• Write down \(a\) and \(d\) clearly before substitution.
• Check whether the sequence is increasing or decreasing.
• Use brackets carefully when multiplying by a negative difference.
• Double-check your final term fits the sequence logic.

Summary

The formula \(u_n = a + (n-1)d\) is the key to finding the \(n\)-th term of an arithmetic sequence. It works by starting from the first term and adding the common difference repeatedly. This simple formula helps you calculate any term, even those far ahead in the sequence, without listing all terms.