Sec 1 Number Patterns: Worked Examples & Practice

Secondary 1 Mathematics · G3 focus · Study guide

Connect term numbers, tables and diagrams to rules for number patterns, including constant differences, decreasing sequences and simple square patterns.

Concept → worked examples → practice14 min read · practice linked below

A term has a position and a value

A sequence is an ordered list of numbers. In $5,8,11,14,\ldots$ , the first term has value 5 and the fourth has value 14. We write $T_n$ for the value at position $n$ . Here positions start at $n=1$ and are positive whole numbers.

A term-to-term rule tells you how to get the next term: “add 3”. A position-to-term rule gives any term directly: $T_n=3n+2$ . The two rules answer different questions. A few numbers alone can fit many rules, so use the stated rule or the structure of a diagram to justify the intended pattern.

Constant differences

If the first term is $a$ and every step adds $d$ , then $T_n=a+(n-1)d.$ There are $n-1$ steps from position 1 to position $n$ . Equivalently, $T_n=dn+(a-d)$ . Check your rule at $n=1$ and at another given position.

Worked example 1: find a distant term

The sequence $5,8,11,14,\ldots$ continues by adding 3. Find its rule and its 20th term.

The first term is 5 and the common difference is $8-5=11-8=3$ .
From the first term to term $n$ , add 3 exactly $n-1$ times: $T_n=5+3(n-1)=\boxed{3n+2}.$
Substitute the position, not a term value: $T_{20}=3(20)+2=\boxed{62}.$

Check: the rule gives $T_1=5$ and $T_4=14$ . It also gives $T_{20}-T_{19}=62-59=3$ , so the difference stays correct.

Explore the starting value and the change

Start with 5 and add 3 each time. The first five terms are 5, 8, 11, 14 and 17. Change the starting value, difference and position; compare a rising, falling and constant sequence.

Equal spacing here represents equal changes in position. The numbers in the boxes give the term values.

Tₙ = 5 + (n − 1) × 3. At n = 5, there are 4 changes, so T₅ = 17.

Decreasing sequences and membership

A negative difference makes a sequence decrease. Zero difference gives a constant sequence. To test whether a value is a term, set $T_n$ equal to that value and solve for $n$ . The answer must be a permitted position: a positive whole number here.

Worked example 2: can a number occur?

The sequence $18,14,10,6,\ldots$ continues by subtracting 4. Is −22 a term? Is −24 a term?

Use $a=18$ and $d=-4$ : $T_n=18-4(n-1)=22-4n.$
For −22, solve $22-4n=-22$ . Then $-4n=-44$ , so $n=11$ . Thus −22 is the 11th term.
For −24, solve $22-4n=-24$ . This gives $n=11.5$ . There is no position 11.5 in the sequence, so −24 is not a term.

Check: $T_{11}=-22$ and $T_{12}=-26$ . The sequence jumps straight from −22 to −26; it does not include every number between them.

Use the structure of a diagram

Look for a part that repeats and a part that stays fixed. A row of joined squares uses three new matchsticks per new square, because the next square shares a side with the previous one.

Each side of one square is one matchstick. Adjacent squares share one upright stick.

Worked example 3: a row of joined squares

Find the number of matchsticks in a row of $n$ joined squares, then decide how many squares can be made with exactly 61 sticks.

There are $n$ top sticks, $n$ bottom sticks and $n+1$ upright sticks.
Add the three groups: $M=n+n+(n+1)=\boxed{3n+1}.$ Equivalently, start with 4 sticks and add 3 for each extra square.
Set $3n+1=61$ . Subtract 1, then divide by 3: $n=\frac{60}{3}=\boxed{20}.$

Exactly 20 joined squares use 61 sticks. Check: $20+20+21=61$ . The rule gives 4 sticks for one square and 10 for the three-square diagram.

Worked example 4: a pattern without a constant difference

Pattern $n$ consists of an $n$ -by- $n$ square of dots and one extra dot outside the square. Find the rule and the seventh term. Can a pattern contain exactly 50 dots?

An $n$ -by- $n$ square has $n\times n=n^2$ dots. Including the extra dot gives $\boxed{T_n=n^2+1}.$
The seventh pattern has $7^2+1=\boxed{50}$ dots.
Conversely, $n^2+1=50$ gives $n^2=49$ . Since a pattern number is positive, $n=7$ , so 50 dots is possible.

Check: the first four values are 2, 5, 10 and 17, with differences 3, 5 and 7. The differences are not constant, so the constant-difference formula does not apply. The square structure explains the rule.

Patterns that multiply instead of add

For some patterns, each term is multiplied by the same number. Write out the multiplications to find how many are needed; do not force a constant-difference rule onto the sequence.

Worked example 5: repeated doubling

A pattern starts with 3 counters and doubles at each stage: $3,6,12,24,\ldots$ . Find a rule and the sixth term.

The first stage has 3 counters. The second has $3\times2$ , and the third has $3\times2\times2=3\times2^2$ .
Reaching stage $n$ requires $n-1$ doublings, so $T_n=3\times2^{n-1}.$
At stage 6, $T_6=3\times2^5=3\times32=\boxed{96}.$

Check by listing: $3,6,12,24,48,96$ . The first term is also correct because $2^0=1$ , so $T_1=3$ . The differences 3, 6 and 12 grow; the repeated multiplier is 2.

Common mistakes and further practice

Using $a+nd$ : position 1 has no change yet, so use $n-1$ changes.
Mixing up position and value: to find the 20th term, put $n=20$ ; to find where 20 occurs, solve $T_n=20$ .
Accepting a fractional or zero position: verify that $n$ is a positive whole number.
Assuming every pattern is linear: check the differences and use the given diagram or rule.

Build confidence with the number patterns and sequences practice bank. Revisit solving linear equations if finding a term's position is difficult.