The common difference is the times table the sequence is built on
The same number of dots at every step
In the sequence 3, 7, 11, 15, … every term is 4 more than the one before. Draw each term as dots and the 4 new dots of each step stand out as a new column.
Each term is the one before with a new column of 4 dots.
The times table underneath
The 4 times table, 4, 8, 12, 16, grows in exactly the same way: 4 more at each step. Its first term is 4 × 1, its second is 4 × 2, and its term in position n is 4 × n, written 4n.
Our sequence is always 1 behind the 4 times table: 3 = 4 − 1, 7 = 8 − 1, 11 = 12 − 1 and 15 = 16 − 1. So its term in position n is 4n − 1. This formula is called the nth term.
The 4 times table is n columns of 4 dots. Our sequence has one dot fewer in every term.
Why the nth term is useful
The nth term gives any term straight from its position, without listing the terms before it. The 100th term is 4 × 100 − 1 = 399. The term-to-term rule, add 4, would need 99 steps to get there.
To check a formula, test it on a term you did not use to find it: the 4th term is 4 × 4 − 1 = 15, which matches.
Finding the nth term
First find the common difference. It is the number that multiplies n. Then write out that times table under the sequence and compare, to find the number to add or subtract.
In 5, 8, 11, 14, … the common difference is 3. The 3 times table is 3, 6, 9, 12, and every term of the sequence is 2 more, so the nth term is 3n + 2. Check: the 3rd term is 3 × 3 + 2 = 11.
Make the fifth term 20
the nth term is a + (n − 1)d, so the common difference is added n − 1 times
Set the first term and the common difference. To reach the fifth term, the common difference is added to the first term 4 times.
The terms lie on a line
Plot each term against its position: (1, 3), (2, 7), (3, 11) and (4, 15). Every step of 1 to the right goes up by 4, so the points lie on a straight line. That is why a sequence with a common difference is called a linear sequence.
The terms 3, 7, 11 and 15 plotted against their positions 1, 2, 3 and 4 rise by 4 at every step, and all four lie on one straight line.
The usual mistakes
The number multiplying n is the common difference, not the first term. 3, 7, 11, 15 has the nth term 4n − 1, not 3n.
Check which way the adjustment goes. The sequence is below the 4 times table, so 1 is subtracted. 4n + 1 would give 5, 9, 13, 17.
Patterns and page numbers · A Row of Matchstick Squares: Figure 45, and the Figure with 241 Sticks
QuestionThe pattern below shows a sequence of squares formed by matchsticks: Figure 1 uses 4 matchsticks to form 1 square. Figure 2 uses 7 matchsticks to form 2 connected squares. Figure 3 uses 10 matchsticks to form 3 connected squares. (a) How many matchsticks are needed to form Figure 45? (b) Which Figure number is made of 241 matchsticks?
1.Figure 1 is the first upright stick and one C of three: 1 + 1 × 3 = 4 sticks.
Figure 1: one first stick, then a C of three: 1 + 1 × 3 = 4. 2.Figure 2 is the same first stick and two Cs: 1 + 2 × 3 = 7 sticks. The second square uses the first square's right side as its left side, so it needs only three new sticks.
Figure 2: the same first stick, then two Cs: 1 + 2 × 3 = 7. 3.Figure 3 is the first stick and three Cs: 1 + 3 × 3 = 10 sticks. So Figure n is the first stick and n Cs.
Figure 3: 1 + 3 × 3 = 10. Every new square is one more C of three. 4.(a) Figure 45 is the first stick and 45 Cs: 1 + 45 × 3 = 136 matchsticks.
(a) Figure 45: 1 + 45 × 3 = 136 sticks. Slide the figure number to see the rule hold. 5.(b) Take the first stick away from 241: 241 − 1 = 240 sticks are in Cs.
(b) 241 sticks: take away the first stick, 241 − 1 = 240. 6.Each C is 3 sticks, so there are 240 ÷ 3 = 80 Cs, one for each square: Figure 80. Check: 1 + 80 × 3 = 241.
240 ÷ 3 = 80 Cs, so Figure 80.
Answer: (a) 136 matchsticks; (b) Figure 80
Watch out for
- Multiplying 45 by 4 because one square takes 4 sticks, which gives 180. Neighboring squares share a side, so after the first square each one needs only 3 new sticks.
- Dividing 241 by 3 at once, which does not divide exactly. The first stick belongs to no C; take it away first: 240 ÷ 3 = 80.
Your turn
1.2, 8, 14, 20, … what is the nth term?