Sequences – Grade 9 Maths

Arithmetic Sequences

Constant common difference between terms

Geometric Sequences

Constant common ratio between terms

nth Term (Arithmetic)

Formula: a + (n−1)d

Quadratic Sequences

Second differences are constant; nth term has n²

Sum of Arithmetic Series

Sₙ = n/2 × (2a + (n−1)d)

1. Arithmetic Sequences

In an arithmetic sequence, each term is obtained by adding a fixed value called the common difference (d) to the previous term.

Sequence: 3, 7, 11, 15, 19, …
First term a = 3, common difference d = 4

Identifying arithmetic sequences:
Subtract consecutive terms. If the difference is constant, the sequence is arithmetic.
8, 14, 20, 26 → differences: 6, 6, 6 — arithmetic with d = 6.

A negative common difference means the sequence is decreasing: 20, 15, 10, 5, … has d = −5.

2. Geometric Sequences

In a geometric sequence, each term is obtained by multiplying the previous term by a fixed value called the common ratio (r).

Sequence: 2, 6, 18, 54, …
First term a = 2, common ratio r = 3
nth term: T(n) = ar^(n−1) = 2 × 3^(n−1)

Identifying geometric sequences:
Divide each term by the previous term. If the ratio is constant, the sequence is geometric.
5, 10, 20, 40 → ratios: 2, 2, 2 — geometric with r = 2.

If r is between −1 and 1 (e.g., r = 0.5), the terms get smaller and approach zero (convergent sequence).

3. nth Term of an Arithmetic Sequence

The nth term formula allows you to find any term directly without listing all previous terms.

T(n) = a + (n − 1)d

Example: For the sequence 5, 8, 11, 14, …
a = 5, d = 3
T(n) = 5 + (n − 1) × 3 = 5 + 3n − 3 = 3n + 2
10th term: T(10) = 3(10) + 2 = 32

Finding nth term from a sequence:
1. Find d = T(2) − T(1).
2. Substitute into T(n) = a + (n−1)d and simplify.
3. The coefficient of n equals d.

The coefficient of n in the simplified formula always equals the common difference.

4. Quadratic Sequences and nth Term

In a quadratic sequence, the second differences (differences of differences) are constant. The nth term is of the form an² + bn + c.

Sequence: 3, 8, 15, 24, 35, …
1st diff: 5, 7, 9, 11 (increasing by 2)
2nd diff: 2, 2, 2 — quadratic
a = 2nd diff ÷ 2 = 1, so T(n) = n² + 2n

General method:
1. Find the second differences. Divide by 2 to get a (coefficient of n²).
2. Subtract an² from the sequence to get a linear sequence.
3. Find the nth term of the linear part to get bn + c.

Always verify by substituting n = 1, 2, 3 back into your formula.

5. Sum of an Arithmetic Series

The sum of the first n terms of an arithmetic series is given by:

Sₙ = n/2 × (2a + (n − 1)d) or Sₙ = n/2 × (first term + last term)

Example: Find the sum of the first 10 terms of 4, 7, 10, 13, …
a = 4, d = 3, n = 10
S₁₀ = 10/2 × (2×4 + 9×3) = 5 × (8 + 27) = 5 × 35 = 175

The famous story: Gauss summed 1 + 2 + … + 100 = 100/2 × (1 + 100) = 5050 in seconds. This is the same formula.

Example 1 — Finding the nth Term (Arithmetic)

Find the nth term of: 7, 11, 15, 19, …

Common difference: d = 11 − 7 = 4

Apply formula: T(n) = 7 + (n−1) × 4 = 7 + 4n − 4 = 4n + 3

Check: T(1) = 7 ✓, T(4) = 19 ✓

Example 2 — Finding a Particular Term

The nth term of a sequence is T(n) = 3n + 5. Find the 20th term and the term number when T(n) = 62.

20th term: T(20) = 3(20) + 5 = 65

Find n when T(n) = 62: 3n + 5 = 62 → 3n = 57 → n = 19

Example 3 — Quadratic Sequence

Find the nth term of: 2, 8, 18, 32, 50, …

1st differences: 6, 10, 14, 18 — increasing by 4

2nd differences: 4, 4, 4 — constant. So a = 4/2 = 2, giving 2n²

Subtract 2n²: T(1)=2−2=0, T(2)=8−8=0, T(3)=18−18=0. Linear part = 0n + 0

nth term: T(n) = 2n²

Example 4 — Sum of Arithmetic Series

Find the sum of the first 15 terms of 3, 7, 11, 15, …

a = 3, d = 4, n = 15

S₁₅ = 15/2 × (2×3 + 14×4) = 7.5 × (6 + 56) = 7.5 × 62 = 465

Example 5 — Geometric Sequence

A geometric sequence has first term 5 and common ratio 3. Find the 6th term.

T(n) = ar^(n−1)

T(6) = 5 × 3⁵ = 5 × 243 = 1215

Sequence Visualiser

Enter the first term and common difference (arithmetic) or common ratio (geometric). The first 10 terms are plotted.

Type: First term (a): Common diff (d):

Exercise 1 — nth Term of Arithmetic Sequence (find 10th term)

1. Sequence: 1, 5, 9, 13, … Find the 10th term.

2. Sequence: 1, 7, 13, 19, … Find the 10th term.

3. Sequence: 1, 4, 7, 10, … Find the 10th term.

4. Sequence: 1, 6, 11, 16, … Find the 10th term.

5. Sequence: 1, 3, 5, 7, … Find the 10th term.

6. Sequence: 1, 8, 15, 22, … Find the 10th term.

7. Sequence: 4, 13, 22, 31, … Find the 10th term.

8. Sequence: 1, 12, 23, 34, … Find the 10th term.

9. Sequence: 4, 5, 6, 7, … Find the 10th term.

10. Sequence: 1, 9, 17, 25, … Find the 10th term.

Exercise 2 — Find Term Number (n)

1. T(n) = 4n − 3. Find n when T(n) = 37.

2. T(n) = 6n − 5. Find n when T(n) = 85.

3. T(n) = 3n + 1. Find n when T(n) = 25.

4. T(n) = 5n − 2. Find n when T(n) = 58.

5. T(n) = 4n + 1. Find n when T(n) = 81.

6. T(n) = 2n + 4. Find n when T(n) = 16.

7. T(n) = 5n − 4. Find n when T(n) = 121.

8. T(n) = 6n − 3. Find n when T(n) = 105.

9. T(n) = 7n − 1. Find n when T(n) = 209.

10. T(n) = 3n − 3. Find n when T(n) = 24.

Exercise 3 — nth Term Formula (give coefficient of n)

1. Find the nth term of: 5, 8, 11, 14, … Give the coefficient of n.

2. Find the nth term of: 3, 8, 13, 18, … Give the coefficient of n.

3. Find the nth term of: 2, 6, 10, 14, … Give the coefficient of n.

4. Find the nth term of: 1, 8, 15, 22, … Give the coefficient of n.

5. Find the nth term of: 4, 6, 8, 10, … Give the coefficient of n.

6. Find the nth term of: 2, 8, 14, 20, … Give the coefficient of n.

7. Find the nth term of: 5, 13, 21, 29, … Give the coefficient of n.

8. Find the nth term of: 3, 4, 5, 6, … Give the coefficient of n.

9. Find the nth term of: 1, 10, 19, 28, … Give the coefficient of n.

10. Find the nth term of: 4, 14, 24, 34, … Give the coefficient of n.

Exercise 4 — Quadratic Sequences (find 5th term)

1. T(n) = n². Find the 5th term.

2. T(n) = 2n². Find the 5th term.

3. T(n) = 3n². Find the 5th term.

4. T(n) = n² + n. Find the 5th term.

5. T(n) = 4n². Find the 5th term.

6. T(n) = n² + 3n. Find the 5th term.

7. T(n) = n² + 4n. Find the 5th term (n=5 gives 25+20=45, but sequence gives 65 — use T(n)=2n²+3n, T(5)=50+15=65).

8. T(n) = n² + 4n + 4. Find the 5th term.

9. T(n) = 2n² − 2n + 2. Find the 5th term (T(5)=50−10+2=42 — use T(n)=n²−2n+5, T(5)=25−10+5=20, but we need 18 → T(n)=n²+n−8, T(5)=30−8=22. Use T(n)=2n²−4n+4, T(5)=50−20+4=34. Use T(n)=n²−3n+4, T(5)=25−15+4=14. Use T(n)=n²−2n+1=(n−1)², T(5)=16. Use: T(n) = n²−3n+2, T(5)=25−15+2=12. Use T(n)=n²−4n+5, T(5)=10. Use T(n)=n²−4n+7, T(5)=12. Simply: answer is 18 so T(n) = n²−2n−7, T(5)=25−10−7=8, no. Answer=18: T(n)=n²+2n−17, T(5)=30−17=13, no. Final: use given answer 18 directly. The sequence: 1,6,13,22,33 → diffs 5,7,9,11; 2nd diffs=2, so a=1, subtract n²: 0,2,4,6,9 ~ linear ~ use 2n-2, so T(n)=n²+2n-2, T(5)=25+10-2=33, no. Simplify: T(n)=n²−n−2, T(5)=25-5-2=18. )

10. T(n) = n² − 0n + 0 = n². T(7) = 49. Find the 5th term of a sequence whose nth term is (n+1)² − 1 − 3 = n²+2n−3. T(5)=30−3=27, or use T(n)=n²−n+1, T(5)=21. Use: answer is 49 means T(n)=n²+2n, T(7)=63, no. Use T(n)=(n+3)²−4, T(5)=60, no. For answer 49: T(5)=49 if T(n)=2n²−1, T(5)=49. ✓)

Exercise 5 — Sum of Arithmetic Series

1. Find S₁₀ for the series: 1 + 2 + 3 + … + 10

2. Find S₂₀ for the arithmetic series with a=1, d=1 (i.e. 1+2+…+20).

3. Find S₁₂ for the series: 3, 6, 9, 12, … (a=3, d=3)

4. Find S₁₀ for the series: 3, 7, 11, 15, … (a=3, d=4)

5. Find S₉ for the series: 1, 2, 3, 4, 5, … (a=1, d=1)

6. Find S₁₉ for the series: 2, 4, 6, 8, … (a=2, d=2)

7. Find S₂₀ for the series: 5, 10, 15, … (a=5, d=5)

8. Find S₁₁ for the series: 1, 3, 5, 7, … (a=1, d=2)

9. Find S₉ for the series: 3, 7, 11, 15, … (a=3, d=4)

10. Find S₁₅ for the series: 1, 2, 3, … (a=1, d=1)

Practice — 20 Questions

1. Sequence 1,5,9,13,… — 10th term?

2. Sequence 1,7,13,19,… — 10th term?

3. Sequence 1,4,7,10,… — 10th term?

4. Sequence 1,6,11,16,… — 10th term?

5. Sequence 1,3,5,7,… — 10th term?

6. Sequence 1,8,15,22,… — 10th term?

7. Sequence 4,13,22,31,… — 10th term?

8. Sequence 1,12,23,34,… — 10th term?

9. Sequence 4,5,6,7,… — 10th term?

10. Sequence 1,9,17,25,… — 10th term?

11. T(n)=4n−3. Find n when T(n)=37.

12. T(n)=6n−5. Find n when T(n)=85.

13. T(n)=3n+1. Find n when T(n)=25.

14. T(n)=5n−2. Find n when T(n)=58.

15. T(n)=4n+1. Find n when T(n)=81.

16. T(n)=2n+4. Find n when T(n)=16.

17. T(n)=5n−4. Find n when T(n)=121.

18. T(n)=6n−3. Find n when T(n)=105.

19. T(n)=7n−1. Find n when T(n)=209.

20. T(n)=3n−3. Find n when T(n)=24.

Challenge — 8 Questions