01 · Write the rule
Find the nth term and 12th term of 4, 9, 14, 19,…
Hint
The first term is 4 and the difference is 5.
Worked solution
uₙ = 4 + 5(n − 1) = 5n − 1. The 12th term is 59.
Understand · explore · practise
Arithmetic sequences
Find the nth term of an arithmetic sequence, recover the first term and common difference, test membership and solve threshold questions with original visual practice.
Before you startSubstitution, linear equations and inequalities
01 / A constant difference
Compare consecutive terms by subtraction.
An arithmetic sequence adds the same number each time. This number is the common difference d. It may be positive, negative or zero.
7, 10, 13, 16, … has d = 3
18, 14, 10, 6, … has d = −4
A constant sequence such as 9, 9, 9, … is arithmetic with d = 0. A few terms alone do not uniquely define an infinite sequence: use the stated arithmetic rule, not an unsupported guess about a pattern.
02 / The nth term
There are n − 1 increments after the first term.
Let a be the first term, u₁. The second term is a + d, the third is a + 2d, and so the nth is a + (n − 1)d, for positive integer n.
uₙ = a + (n − 1)d
7, 10, 13, 16, …Worked example
uₙ = 7 + 3(n − 1) = 3n + 4
Check n = 1 gives the first term 7.
u₂₀ = 7 + 19 × 3 = 64
The 20th term is reached after 19 increments.
u₁ = 3(1) + 4 = 7
The constant 4 in 3n + 4 is not the first term.
Count the steps from the first termExplore
Each dot is a term at an integer index. The first term stays at 5. The highlighted dot is the selected term.
u₄ = 5 + 3 × 2 = 11. Four terms contain three increments.
Positive difference: the sequence increases.
Watch why the formula uses n − 1
Pause, replay or seek freely. The notes explain the same idea and stay in view.
03 / Use two known terms
The index gap tells you how many equal increments lie between them.
Given u₄ = 17 and u₁₁ = 45Worked example
u₁₁ − u₄ = (11 − 4)d = 7d
Subtracting the two term equations cancels a.
d = (45 − 17)/7 = 4
There are 7 increments between the 4th and 11th terms.
a + 3d = 17 ⇒ a = 5
The 4th term has 3 increments after the first.
uₙ = 5 + 4(n − 1) = 4n + 1
Substitution checks both supplied terms.
More generally, u_q − u_p = (q − p)d. The indices p and q must differ to determine d this way.
04 / Is a number in the sequence?
The index must be a positive integer.
Does the sequence uₙ = 3n + 4 contain 85 or 86?Worked example
3n + 4 = 85 ⇒ n = 27
Yes: 85 is the 27th term.
3n + 4 = 86 ⇒ n = 82/3
No: a non-integer index does not identify a term.
3n + 4 = 1 ⇒ n = −1
This also fails: indices start at 1 here.
When d = 0, do not divide by d. The sequence is constant: its repeated value occurs at every positive index and every other number occurs at none.
05 / The first term crossing a boundary
Solve the inequality, then test neighbouring integer indices.
First negative term when a = 42 and d = −5Worked example
uₙ = 42 − 5(n − 1) = 47 − 5n
The sequence is decreasing.
47 − 5n < 0 ⇒ n > 47/5 = 9.4
Dividing by −5 reverses the inequality.
First possible integer n = 10
Check u₉ = 2 and u₁₀ = −3. Therefore −3 is the first negative term.
Strict and non-strict inequalities can give different answers. For uₙ = 24 − 3(n − 1), the first nonpositive term is u₉ = 0, while the first negative term is u₁₀ = −3.
Check whether a boundary can be reached at all. A positive increasing sequence cannot later cross below zero. Do not round an index without considering the direction and strictness of the inequality.
06 / Unknown parameters
Set adjacent differences equal.
k − 1, 2k + 3, 4k + 1 are consecutive arithmetic termsWorked example
Second − first = k + 4
Keep the brackets while subtracting.
Third − second = 2k − 2
Both differences must agree.
k + 4 = 2k − 2 ⇒ k = 6
The terms are 5, 15, 25, with common difference 10.
For three consecutive terms A, B, C, the same condition can be written 2B = A + C. If the resulting parameter equation has several roots, check all of them against any stated restrictions.
07 / Symbolic and logarithmic terms
A common difference need not be an integer.
The sequence ln 5, ln 20, ln 80, … is arithmetic because ln 20 − ln 5 = ln 4 and ln 80 − ln 20 = ln 4. Its nth term is ln 5 + (n − 1)ln 4, equivalently ln(5 × 4^(n − 1)).
The numbers inside the logarithms form a geometric sequence, while their logarithms form an arithmetic sequence. Check that all logarithm arguments are positive before using the log laws.
For a symbolic example, t, t + h, t + 2h, … has first term t and common difference h. The nth term is t + (n − 1)h; no numerical substitution is needed.
08 / Your turn
Keep the first term and the number of increments distinct.
Assume every listed sequence is stated to be arithmetic. Show an integer-index check where needed.
02 · Decreasing sequence
Find the nth term of 31, 25, 19, 13,… and its 8th term.
Hint
The common difference is negative.
Worked solution
uₙ = 31 − 6(n − 1) = 37 − 6n. Thus u₈ = −11.
03 · Two known terms
u₃ = 14 and u₉ = 38. Find a, d and uₙ.
Hint
The two terms are 6 increments apart.
Worked solution
d = (38 − 14)/6 = 4. Since a + 2d = 14, a = 6. Therefore uₙ = 4n + 2.
04 · Membership
For uₙ = 5n − 1, decide whether 74 and 75 occur.
Hint
Solve each term equation for n.
Worked solution
74 gives n = 15, so it occurs. 75 gives n = 76/5, not an integer, so it does not.
05 · Count a finite list
How many terms are in 8, 13, 18,…,123?
Hint
Set the nth-term formula equal to the final value.
Worked solution
8 + 5(n − 1) = 123 gives n − 1 = 23, so there are 24 terms.
06 · First below zero
For uₙ = 50 − 4(n − 1), find the first negative term and its index.
Hint
Solve 54 − 4n < 0 and check the neighbouring indices.
Worked solution
n > 13.5, so the first is u₁₄ = −2. The preceding term u₁₃ = 2 confirms the boundary.
07 · A parameter
3p, p + 8 and p + 14 are consecutive terms of an arithmetic sequence. Find p and the common difference.
Hint
Equate the two differences.
Worked solution
8 − 2p = 6 gives p = 1. The terms are 3, 9, 15, so d = 6.
08 · A constant sequence
An arithmetic sequence has u₂ = 7 and u₆ = 7. Find u₁₀₀.
Hint
Use the difference of the two known terms.
Worked solution
4d = 0, so d = 0 and every term is 7. Hence u₁₀₀ = 7.
09 · Logarithmic terms
Find the common difference and nth term of ln 3, ln 15, ln 75,…
Hint
Subtract logarithms by taking the logarithm of a ratio.
Worked solution
d = ln 5 and uₙ = ln 3 + (n − 1)ln 5 = ln(3 × 5^(n − 1)). All arguments are positive.
10 · Strict boundary
uₙ = 30 − 5(n − 1). Find the first nonpositive term and the first negative term.
Hint
Check where the sequence equals zero.
Worked solution
u₇ = 0 is the first nonpositive term. u₈ = −5 is the first negative term. The difference comes from ≤ versus <.
09 / Recap
An nth-term rule needs an index convention.
- Arithmetic means constant consecutive difference.
- With u₁ = a, use uₙ = a + (n − 1)d.
- Two known terms determine d through their index gap.
- Membership needs a positive integer index.
- Check adjacent integer terms for strict threshold questions.
Section 1 of 9 · A constant difference