Algebraic expressions & sequences · Ages 12–13
Growing patterns and the n-th term
From term 1 to term n, only n − 1 additions.
How to do it
Method
From term 1 to term n, only n − 1 additions
3 + (n − 1) × 4 = 4n − 1
Another way
Compare with multiples
4, 8, 12, 16 each less 1: 3, 7, 11, 15, so 4n − 1
In full
When the same amount is added each time, the n-th term = first term + (n − 1) × the amount added. From term 1 to term n the amount is added only n − 1 times, not n: the commonest slip. For 3, 7, 11, 15, the n-th term = 3 + (n − 1) × 4 = 4n − 1. Or compare with multiples: adding 4 each time, compare with 4n (4, 8, 12, 16). Every term is 1 less, so the n-th term is 4n − 1. For a dot pattern, count each picture's dots first, write them as a sequence, then look for the rule. If the amount added keeps growing by one, add step by step (1, 3, 6, 10: the n-th is 1 + 2 + … + n); don't treat it as the same amount each time. Then put n = 1 and 2 back in to check.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
The n-th term of a sequence is 9n + 4. Term 19 is .
Show the answer and working
Answer: 175
Put 19 for n: 9 × 19 + 4 = 171 + 4 = 175.
Example 2
Count the dots in each pattern. How many dots are in pattern 9?
- A. 48
- B. 27
- C. 45
- D. 43
Choose the right answer
Show the answer and working
Answer: D. 43
The dots go 3, 8, 13, adding 5 each time. From pattern 1 to pattern 9 that is 8 additions: 3 + 8 × 5 = 3 + 40 = 43.
Example 3
The n-th term of a sequence is 11n + 19. 162 is term .
Show the answer and working
Answer: 13
Backwards: 162 − 19 = 143, 143 ÷ 11 = 13.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The n-th term of a sequence is 4n + 1. Term 18 is .
Answer:
2.Count the matchsticks in each pattern. How many are in pattern n?
- A. 5n + 1
- B. 5n + 6
- C. 6n
- D. n + 5
Choose the right answer
Answer:
3.The n-th term of a sequence is 9n + 19. 316 is term .
Answer:
4.The n-th term of 19, 26, 33, 40, … can be written an + b: a = , b = .
Fill in each blank
Answer:
5.The n-th term of 4, 7, 12, 19, … is n2 + b, where b = .
Answer:
All the answers
- 1. 73
- 2. A. 5n + 1
- 3. 33
- 4. 7, 12
- 5. 3
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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