Sequences can go up (increasing) or down (decreasing). The difference between consecutive terms tells us the pattern.

1. Increasing Sequences

In an increasing sequence, each term is larger than the one before it. We add the same number each time.

Example 1: 1, 4, 7, 10, 13, 16, 19, __, __

Look at how the numbers change:

• 1 → 4: add 3
• 4 → 7: add 3
• 7 → 10: add 3
• 10 → 13: add 3

The rule is: add 3 each step. So:

• 19 + 3 = 22
• 22 + 3 = 25

The complete sequence is: 1, 4, 7, 10, 13, 16, 19, 22, 25

Example 2: 1, 3, 5, 7, 9, ___, ___

• 1 → 3: add 2
• 3 → 5: add 2
• 5 → 7: add 2

The rule is: add 2 each step. So:

• 9 + 2 = 11
• 11 + 2 = 13

The complete sequence is: 1, 3, 5, 7, 9, 11, 13

2. Decreasing Sequences

In a decreasing sequence, each term is smaller than the one before it. We subtract the same number each time.

Example 3: 25, 23, 21, __, __

• 25 → 23: subtract 2
• 23 → 21: subtract 2

The rule is: subtract 2 each step. So:

• 21 − 2 = 19
• 19 − 2 = 17

The complete sequence is: 25, 23, 21, 19, 17

Example 4: 45, 41, 37, 33, ____

• 45 → 41: subtract 4
• 41 → 37: subtract 4
• 37 → 33: subtract 4

The rule is: subtract 4 each step. So:

• 33 − 4 = 29

The complete sequence is: 45, 41, 37, 33, 29

1. Look at the first two numbers and find the difference between them.
2. Check if the same difference works for the next pair of numbers.
3. Apply the rule (add or subtract) to find the missing numbers.
4. Write the complete sequence with all terms.

Example A: Find the next three numbers

5, 10, 15, 20, __, __, __

• 5 → 10: add 5
• 10 → 15: add 5
• 15 → 20: add 5

Rule: add 5 each step

• 20 + 5 = 25
• 25 + 5 = 30
• 30 + 5 = 35

Answer: 5, 10, 15, 20, 25, 30, 35

Example B: Find the missing number

50, 45, 40, 35, ___

• 50 → 45: subtract 5
• 45 → 40: subtract 5
• 40 → 35: subtract 5

Rule: subtract 5 each step

• 35 − 5 = 30

Answer: 50, 45, 40, 35, 30

Example C: Larger steps

100, 90, 80, 70, ___, ___

• 100 → 90: subtract 10
• 90 → 80: subtract 10
• 80 → 70: subtract 10

Rule: subtract 10 each step

• 70 − 10 = 60
• 60 − 10 = 50

Answer: 100, 90, 80, 70, 60, 50

• Sequence: a list of numbers in a special order
• Term: each number in a sequence
• Pattern: the rule that tells us how the numbers change
• Increase: when numbers get bigger (add)
• Decrease: when numbers get smaller (subtract)

In Tanzania, sequences are used in many everyday situations. For example, a shopkeeper at the maduka ya duka may arrange prices in a sequence, like TZS 500, TZS 1,000, TZS 1,500, TZS 2,000 — increasing by TZS 500 each time. Similarly, a farmer counting seedlings planted in rows that increase by the same number each day uses sequences to plan and organize their work efficiently.