Problem solving with non-linear relationships | Maths

Starter quiz

Which of these are triangular numbers?
- 0
- 1 ✓
- 5
- 10 ✓
- 20
Use a calculator to work out the value of $3^{8}$ . ______.
- '6561' ✓
The next term in the geometric sequence 0.4, 2, 10, ... is ______.
- '50' ✓
Which of these could be the first 4 terms in an arithmetic sequence?
- 28, 22, 16, 11, ...
- 136, 163, 190, 209, ...
- 13, 49, 85, 121, ... ✓
- 17, 9, 1, -7, ... ✓
- -21, -8, 8, 21, ...
What is the $n^{\text{th}}$ term rule for the linear sequence which starts 13, 8, 3, -2, ...?
- $18 - 5n$ ✓
- $13 - 5n$
- $8 - 5n$
- $5n + 8$
- $5n + 13$
Which of these could be the first 5 terms of a sequence with a common second difference? (These are called quadratic sequences).
- 2, 3, 5, 8, 12, ... ✓
- 8, 12, 18, 28, 40, ...
- 9, 10, 12, 16, 24, ...
- 12, 17, 27, 42, 62, ... ✓
- 15, 21, 28, 36, 45, ... ✓

Exit quiz

This table shows the value of a car two years after it was made. If the values form a geometric sequence, what is the common multiplier between terms? (You may use your calculator.)
- 0.2
- 0.4
- 0.6
- 0.8 ✓
- 1.25
This table shows the value of a car 2 years after it was made. If the values continue to follow the same geometric sequence, what is the car's value after 3 years? (You may use your calculator.) £______
- '10 240' ✓
This table shows the value of a car two years after it was made. Why might this sequence not continue as the same geometric sequence forever?
- If the geometric sequence continues the value will drop below zero pounds.
- Sequences cannot go on forever, they will eventually stop.
- The value of the car may stop decreasing, it could reach minimum value. ✓
- The value may decrease sharply in the first 2 years, then the pattern may change ✓
- The car loses the most value in the first year, not the same amount each year.
Which two sequences can be added to make the first 5 terms of a linear sequence?
- 3, 7, 11, 15, 19, ... ✓
- 5, 10, 20, 40, 80, ...
- 6, 7, 9, 12, 16, ...
- 14, 11, 8, 5, 2, .. ✓
- 16, 24, 36, 54, 81, ...
Order these terms so they form part of a Collatz sequence: "If the term is even, the next term is half the previous term. If the term is odd, the next term is 3 times the previous term add 1".
- 1
  
  ⇔
  
  33
- 2
  
  ⇔
  
  100
- 3
  
  ⇔
  
  50
- 4
  
  ⇔
  
  25
- 5
  
  ⇔
  
  76
- 6
  
  ⇔
  
  38
Which two sequences can be added to make the first 5 terms of a linear sequence?
- 1, 3, 8, 16, 27, ... ✓
- 8, 11, 13, 14, 14, ...
- 10, 19, 25, 28, 28, ... ✓
- 15, 16, 19, 24, 31, ...
- 20, 17, 14, 11, 8, ...

Worksheet

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Lesson Details

Key learning points

If you can spot a sequence, it is possible to predict behaviour.
By predicting future behaviour, you can plan how to react.
As with all predictions, it is not a guarantee.

Common misconception

If you combine two sequences of the same type the resulting sequence will still be that type.

Adding corresponding terms of two sequences will be good preparation for future units but it also a way to explore what sequences can be generated by combining other sequences.

Keywords

Geometric sequence - A geometric sequence is a sequence with a constant multiplicative relationship between successive terms.
Triangular number - A triangular number (or triangle number) is a number that can be represented by a pattern of dots arranged into an equilateral triangle.