Arithmetic Sequences: nth Term, Common Difference and UPCAT Practice
Arithmetic Sequences
Recognize constant differences, move efficiently between terms, and translate sequence information into equations.
Arithmetic Sequences
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Arithmetic sequences change by a constant difference
An arithmetic sequence is an ordered list of numbers in which the same amount is added or subtracted each time. That fixed amount is the common difference, written as d.
For example, in 8, 13, 18, 23, ..., every term increases by 5, so d=5.
aₙ = a₁ + (n−1)d
aₙ is the term you want, a₁ is the first term, n is its position, and d is the common difference. The expression n−1 counts how many jumps occur after the first term.
Find the 4th term of 8, 13, 18, 23, ...
a₄ = 8 + (4−1)(5) = 8 + 15 = 23.
This matches the visible jumps: 8 → 13 → 18 → 23. Reaching the 4th term requires three jumps of 5.
Common difference
Subtract consecutive terms in the same order: later term minus earlier term.
nth term
The first term needs zero jumps; the nth term needs n−1 jumps.
Count jumps, not term numbers
Why it works
Each jump adds d exactly once. The number of additions is the difference between the two term positions.
Five forms you should recognize
7,11,15,... has d=4.
a₁=8,d=5: a₁₅=8+14(5)=78.
5,__,13 has equal gaps, so the middle term is 9.
a₄=14 and a₉=34: five jumps total 20, so d=4.
A theater row with 18 seats, then 22, then 26 follows aₙ=18+4(n−1).
Check before you commit
- Using n instead of n−1: the first term has made zero jumps.
- Changing differences: verify at least two consecutive gaps.
- Wrong subtraction order: later minus earlier keeps the sign meaningful.
- Confusing term and value: “which term is 50?” asks for n.
- Assuming arithmetic: constant ratios indicate a geometric sequence instead.
- Ignoring position gaps: a₉−a₄ covers five, not nine, jumps.
Do you need the lesson-or just practice?
One original question in each form recommends your next step. It does not yet verify mastery.
Work at the level you need.
Foundations
Build the core procedure with immediate explanations.
Core Practice
Use mixed forms with less scaffolding.
UPCAT-Style Transfer
Apply the competency in unfamiliar representations.
Ready to verify this competency?
A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.
Arithmetic Sequences FAQ
How do I know a sequence is arithmetic?
Its consecutive differences are constant.
Why is the formula n−1?
The first term is already present, so reaching term n requires n−1 additions.
Can d be negative?
Yes. A decreasing arithmetic sequence has a negative common difference.
How do I find which term equals a value?
Set the nth-term expression equal to that value and solve for n.
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