🔢 Sequences & nth Term

Find the rule, write the formula, and predict any term in a sequence.

nth Term Formula
nth term = d×n + c
where d = common difference, c = zero term
Finding c (zero term)
c = first term − d
e.g. sequence 5, 8, 11: d=3, c=5−3=2 → 3n+2
Is a number in the sequence?
Set nth term = the number, solve for n
If n is a whole number → YES ✓ | If not → NO ✗

⭐ Golden Rules

  • Common difference d = any term − the previous term
  • nth term = dn + c where c = first term − d
  • To find if X is in the sequence: solve dn + c = X → check if n is an integer
  • Geometric sequences: multiply by the same ratio each time
  • Square numbers: 1, 4, 9, 16, 25 …  |  Triangular: 1, 3, 6, 10, 15 …
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📖 Learn: Sequences & nth Term

What is a sequence?

A sequence is an ordered list of numbers that follow a rule. Each number is called a term.

3,  7,  11,  15,  19,  …

The term-to-term rule tells you what to do to get the next term. Here: add 4.

Arithmetic Sequences — Common Difference

In an arithmetic sequence, you add or subtract the same value each time. That value is called the common difference (d).

d = any term − the term before it
Sequence: 5, 8, 11, 14 → d = 8 − 5 = 3
Sequence: 20, 15, 10, 5 → d = 15 − 20 = −5 (decreasing)

Finding the nth Term Formula

The nth term is a formula that gives you any term in the sequence when you substitute a value of n.

📐 nth term = d × n + c

To find c (called the zero term — the value before the 1st term when n = 0):

c = first term − d
Then substitute to check: when n = 1, does the formula give the 1st term?
Example: Sequence 5, 8, 11, 14
d = 3  |  c = 5 − 3 = 2  |  nth term = 3n + 2
Check: n=1 → 3(1)+2 = 5 ✓   n=2 → 3(2)+2 = 8 ✓

Using the nth Term

Find the 10th term of 3n + 2: substitute n = 10 → 3(10) + 2 = 32
Find the 100th term: n = 100 → 3(100) + 2 = 302
Is 50 in the sequence? 3n + 2 = 50 → 3n = 48 → n = 16 → Integer ✓ → YES
Is 45 in the sequence? 3n + 2 = 45 → 3n = 43 → n = 14.3… → Not integer → NO

Geometric Sequences

In a geometric sequence, you multiply by the same value each time — the common ratio (r).

2, 6, 18, 54, … → multiply by 3 each time (r = 3)
100, 50, 25, 12.5, … → multiply by 0.5 each time (r = 0.5 = ½)
To find r: divide any term by the term before it  →  r = term₂ ÷ term₁

Special Sequences

Square numbers: 1, 4, 9, 16, 25, 36, … (nth term = n²)
Triangular numbers: 1, 3, 6, 10, 15, 21, … (add 2, then 3, then 4, …)
Fibonacci: 1, 1, 2, 3, 5, 8, 13, … (add the two previous terms)

⭐ Golden Rules — Quick Reference

  • d = (any term) − (previous term) — must be the same for ALL pairs
  • c = first term − d  (the "zero term")
  • nth term = dn + c → always check by substituting n = 1 and n = 2
  • If the answer to "solve nth term = value" is an integer, the value IS in the sequence
  • Geometric: multiply (or divide) by the same ratio each time
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💡 Worked Examples

Example 1 — Find the nth term of 4, 7, 10, 13

Step 1: Find d  →  d = 7 − 4 = 3
Step 2: Find c (zero term)  →  c = 4 − 3 = 1
Step 3: Write formula  →  nth term = 3n + 1
✅ Check: n=1 → 3+1=4 ✓   n=2 → 6+1=7 ✓   n=3 → 9+1=10 ✓

Example 2 — Find the nth term of 2, 7, 12, 17

Step 1: d = 7 − 2 = 5
Step 2: c = 2 − 5 = −3
Step 3: nth term = 5n − 3
✅ Check: n=1 → 5−3=2 ✓   n=4 → 20−3=17 ✓

Example 3 — Is 58 in the sequence 4n − 2?

Set formula equal to 58:   4n − 2 = 58
Add 2 to both sides:   4n = 60
Divide by 4:   n = 15
n = 15 is a whole number → YES, 58 is in the sequence (it's the 15th term)

Example 4 — Is 47 in the sequence 3n + 1?

Set formula equal to 47:   3n + 1 = 47
Subtract 1:   3n = 46
Divide by 3:   n = 15.33…
n is not a whole number → NO, 47 is NOT in the sequence
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🔢 nth Term Builder

Enter the first 4 terms of an arithmetic sequence. The tool will find the nth term formula with full working.

📐 Working

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🎯 Exercise 1 — Term-to-Term Rule

Find the common difference for each sequence. Type a number (e.g. 4 for "add 4", −3 for "subtract 3").

3, 7, 11, 15, 19
Rule (add/subtract):
25, 20, 15, 10, 5
Rule:
1, 4, 7, 10, 13
Rule:
50, 43, 36, 29, 22
Rule:
−3, 2, 7, 12, 17
Rule:
8, 11, 14, 17, 20
Rule:
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🎯 Exercise 2 — Fill the Missing Terms

Complete each arithmetic sequence by finding the two missing terms.

3, , 11, , 19
5, 8, , 14,
20, , 10, , 0
−3, , 5, , 13
2, , , 11, 14
1, 6, , 16,
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🎯 Exercise 3 — Match nth Term to Sequence

Click an nth term formula, then click the sequence it produces. Match all 4 pairs!

2n + 1
3n − 2
5n
4n + 3
5, 10, 15, 20, 25
3, 5, 7, 9, 11
7, 11, 15, 19, 23
1, 4, 7, 10, 13
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🎯 Exercise 4 — Find the nth Term (Step by Step)

For each sequence, find d (common difference), the zero term c (= first term − d), and the nth term formula.

5, 8, 11, 14
2, 7, 12, 17
10, 13, 16, 19
1, 5, 9, 13
6, 10, 14, 18
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🎯 Exercise 5 — Is It in the Sequence?

For each question, solve the equation to find n. If n is a whole number, the value IS in the sequence. Click YES or NO.

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🎯 Exercise 6 — Geometric Sequences

Find the common ratio r (multiply by what each time?) and write the next two terms.

2, 4, 8, 16, ___, ___
3, 9, 27, ___, ___
5, 10, 20, ___, ___
2, 6, 18, ___, ___
4, 20, 100, ___, ___
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✏️ Practice Quiz

20 questions covering all topics. Type your answers and click Check Answers.

  1. Common difference in 4, 7, 10, 13, … =
  2. Next term: 2, 5, 8, 11, 14, ___ =
  3. Next term: 30, 24, 18, 12, ___ =
  4. nth term of 3, 5, 7, 9 =
  5. nth term of 4, 7, 10, 13 =
  6. nth term of 1, 6, 11, 16 =
  7. nth term of 8, 11, 14, 17 =
  8. 10th term if nth term = 4n + 3 =
  9. 15th term if nth term = 2n − 1 =
  10. 100th term if nth term = 3n + 2 =
  11. Is 44 in the sequence 5n + 3? (YES or NO)
  12. Is 50 in the sequence 4n + 2? (YES or NO)
  13. Is 51 in the sequence 7n − 2? (YES or NO)
  14. Common difference of −5, −1, 3, 7 =
  15. Next two terms of 6, 10, 14, 18, ___, ___: first blank = , second blank =
  16. nth term of 6, 11, 16, 21 =
  17. 5th term of nth term = 6n − 3 =
  18. Common ratio of 2, 6, 18, 54 =
  19. Next term of 4, 12, 36, ___ =
  20. Is 40 in the sequence 4n − 4? (YES or NO)
Practice Answers: 1. 3  |  2. 17  |  3. 6  |  4. 2n+1  |  5. 3n+1  |  6. 5n−4  |  7. 3n+5  |  8. 43  |  9. 29  |  10. 302  |  11. NO (5n+3=44→5n=41→n=8.2)  |  12. YES (4n+2=50→4n=48→n=12)  |  13. NO (7n−2=51→7n=53→n=7.57)  |  14. 4  |  15. 22, 26  |  16. —  |  17. 5n+1  |  18. 27  |  19. 3  |  20. 108  |  21. YES (4×11−4=40)
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🏆 Challenge Questions

10 harder questions. Show your working on paper, then type the answers.

  1. The nth term of a sequence is 3n + k. The 5th term is 23. Find k.
  2. Which term of the sequence 4n − 1 equals 79? (find n)
  3. Sequences: 2n + 5 and 4n − 3. Find the value of n where they are equal.
  4. The value they share (from Q3) is:
  5. The first term of a sequence is 7 and the 4th term is 19. Find the nth term formula.
  6. nth term = 5n + c. The 3rd term is 18. Find c.
  7. Then find the 10th term of this sequence (from Q6).
  8. Sequence: 2, ___, ___, 14 (arithmetic). Find the two missing terms. First: Second:
  9. Geometric: first term = 3, ratio = 4. Find the 5th term.
  10. How many terms in the sequence 3n + 1 are less than 100?
Challenge Answers:
1. k=8 (15+k=23)  |  2. n=20 (4n=80)  |  3. n=4 (2n+5=4n−3 → 2n=8)  |  4. 13  |  5. 4n+3 (d=4, c=7−4=3)  |  6. c=3 (15+c=18)  |  7. 53 (5×10+3)  |  8. 6, 10 (d=4)  |  9. 768 (3×4⁴=3×256)  |  10. 32 terms (3n+1<100 → n<33)
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