Patterns & Sequences • Topic 1 of 3

Number Patterns

What are Number Patterns?

A number pattern is a sequence of numbers that follows a specific rule or relationship. Each number in the pattern is called a term. Patterns help us predict future terms and understand mathematical relationships.

Real-life Example: Saving money each week - if you save ₹100 in week 1, ₹200 in week 2, ₹300 in week 3, the pattern shows you save ₹100 more each week.

What are Arithmetic Sequences?

An arithmetic sequence (or arithmetic progression) is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference ($d$).

Formula for Arithmetic Sequence:

$a_n = a_1 + (n-1)d$

Where:

$a_n$ = the $n$th term
$a_1$ = first term
$d$ = common difference
$n$ = position of the term

Examples of Arithmetic Sequences:

Sequence	First Term ($a_1$)	Common Difference ($d$)	Next Term
2, 5, 8, 11, ...	2	+3	14
20, 15, 10, 5, ...	20	-5	0
-3, -1, 1, 3, ...	-3	+2	5
7, 7, 7, 7, ...	7	0	7

Key Vocabulary:

Term: Each number in the sequence (denoted as $a_1, a_2, a_3, ...$)
Common Difference: The fixed number added (or subtracted) to get from one term to the next
Finite Sequence: Has a limited number of terms
Infinite Sequence: Continues forever (indicated by ...)

Solved Examples

Worked Example

Example 1: Identify if the sequence 7, 12, 17, 22, 27 is an arithmetic sequence. If yes, find the common difference and the next two terms.

Solution- Step 1: Find difference between consecutive terms - Step 2: $12 - 7 = 5$ - Step 3: $17 - 12 = 5$ - Step 4: $22 - 17 = 5$ - Step 5: $27 - 22 = 5$ - Step 6: All differences are equal to 5, so it is an arithmetic sequence - Step 7: Common difference $d = 5$ - Step 8: Next term = $27 + 5 = 32$ - Step 9: Second next term = $32 + 5 = 37$

Worked Example

Example 2: Find the 15th term of the arithmetic sequence: 10, 7, 4, 1, ...

Solution- Step 1: Identify $a_1 = 10$ - Step 2: Find common difference: $d = 7 - 10 = -3$ - Step 3: Use formula $a_n = a_1 + (n-1)d$ - Step 4: For $n = 15$: $a_{15} = 10 + (15-1)(-3)$ - Step 5: $a_{15} = 10 + 14 \times (-3)$ - Step 6: $a_{15} = 10 - 42$ - Step 7: $a_{15} = -32$

Worked Example

Example 3: In an arithmetic sequence, the 5th term is 22 and the 9th term is 38. Find the first term and the common difference.

Solution- Step 1: Write given information: $a_5 = 22$, $a_9 = 38$ - Step 2: Use formula: $a_5 = a_1 + 4d = 22$ ...(1) - Step 3: $a_9 = a_1 + 8d = 38$ ...(2) - Step 4: Subtract (1) from (2): $(a_1 + 8d) - (a_1 + 4d) = 38 - 22$ - Step 5: $4d = 16$ - Step 6: $d = 4$ - Step 7: Substitute $d = 4$ in equation (1): $a_1 + 4(4) = 22$ - Step 8: $a_1 + 16 = 22$ - Step 9: $a_1 = 6$

Key Points

An arithmetic sequence has a constant difference between consecutive terms
Common difference $d = a_2 - a_1 = a_3 - a_2 = ...$
Formula for nth term: $a_n = a_1 + (n-1)d$
$d$ can be positive (increasing sequence), negative (decreasing sequence), or zero (constant sequence)
To find missing terms, use the formula with given information
Arithmetic sequences appear in real life: savings, timetables, distances, etc.
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Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.The next term of $2, 4, 6, 8, \ldots$ is:

Explanation: Add $2$.

Q2.The numbers $1, 4, 9, 16$ are:

Explanation: $1^2,2^2,3^2,4^2$.

Q3.The rule for $5, 10, 15, 20$ is:

Explanation: Add $5$.

Q4.The next term of $2, 6, 18, 54$ is:

Explanation: Multiply by $3$.

Practice more MCQs → 📝 Assignment · 35 marks