Patterns & Sequences — Grade 5 Mathematics

In Grade 5, we work with arithmetic sequences that may have fractional or negative common differences, and we are introduced to geometric sequences where each term is multiplied by a fixed ratio.

Arithmetic Sequences — nth Term

An arithmetic sequence has a fixed common difference (d) between terms.

nth term formula: T(n) = a + (n − 1) × d
where a = first term, d = common difference.

Example: 3, 7, 11, 15, … here a = 3, d = 4 → T(n) = 3 + (n−1) × 4 = 4n − 1

Fractional / Negative Differences

−1.5

8.5

−1.5

5.5

a = 10, d = −1.5 → T(n) = 10 + (n−1) × (−1.5) = 11.5 − 1.5n

Geometric Sequences

A geometric sequence multiplies each term by a fixed common ratio (r).

×3

First term = 2, common ratio = 3. The 5th term = 54 × 3 = 162.

💡 Geometric sequences can grow very rapidly or shrink toward zero (when 0 < r < 1).

Worked Examples

Example 1 — Find the nth term

Find the nth term of: 5, 8, 11, 14, …

d = 8 − 5 = 3; a = 5
T(n) = 5 + (n−1) × 3 = 5 + 3n − 3 = 3n + 2
Check: T(1) = 3(1) + 2 = 5 ✓

Example 2 — Is 50 in the sequence?

Sequence: 3n + 2. Is 50 a term?

Set 3n + 2 = 50 → 3n = 48 → n = 16
n = 16 is a whole number, so yes — it is the 16th term.

Example 3 — Geometric Sequence

A sequence is 4, 12, 36, 108, … Find the next term and the 6th term.

Common ratio r = 12 ÷ 4 = 3
5th term: 108 × 3 = 324
6th term: 324 × 3 = 972

Practice Questions

1. Find the nth term of: 7, 12, 17, 22, …

✅ d = 5, a = 7 → T(n) = 7 + (n−1)×5 = 5n + 2

2. Find the 10th term of the sequence with nth term 4n − 3.

✅ T(10) = 4(10) − 3 = 40 − 3 = 37

3. A sequence is: 20, 17.5, 15, 12.5, … Write the nth term.

✅ d = −2.5, a = 20 → T(n) = 20 + (n−1)(−2.5) = 22.5 − 2.5n

4. Is 41 a term in the sequence 3n + 2?

✅ 3n + 2 = 41 → 3n = 39 → n = 13. Yes — it is the 13th term.

5. A geometric sequence starts 5, 10, 20, 40, … Find the 6th term.

✅ r = 2; terms: 5, 10, 20, 40, 80, 160

Key Points to Remember

Arithmetic sequence: add (or subtract) the same value each time — this is the common difference d.
nth term formula: T(n) = a + (n − 1) × d; simplify to the form dn + c.
To check if a value is in a sequence, solve T(n) = value. If n is a positive integer, it's in the sequence.
Geometric sequence: multiply by the same value each time — this is the common ratio r.
Geometric sequences with r > 1 grow fast; with 0 < r < 1 they decrease toward zero.

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