Distinguish arithmetic and geometric patterns

The displayed terms are 10, 15, 20, 25. Which description fits every pair?

Start with 8 and change the amount added each step. Compare adding 4 with adding 0. Check the displayed values, not just the shape of the plot

When the added amount is 0, which patterns fit?

Two separate tests

Do not start by assuming that a sequence is geometric. Test the displayed values for repeated addition and repeated multiplication

1. 1Test addition Subtract each earlier value from the next. Equal differences across every displayed pair fit an arithmetic pattern
2. 2Test multiplication Divide each later value by the earlier value. Equal ratios across every displayed pair fit a geometric pattern. The terms here are positive, so these divisions are defined

A geometric pattern can decrease

For 64, 48, 36, 27, the ratio is three quarters at every step. The differences are not constant

1. 1Ratios Compare later value divided by earlier value \(\displaystyle \frac{48}{64}=\frac{36}{48}=\frac{27}{36}=\frac34\)
2. 2Differences The successive changes are different \(\displaystyle 48-64=-16,\quad36-48=-12,\quad27-36=-9\)

Constant fits both

For 9, 9, 9, 9, every difference is zero and every ratio is one. Neither definition requires the values to change

1. 1Arithmetic test Adding zero preserves the value \(\displaystyle 9-9=0\)
2. 2Geometric test Multiplying by one preserves the value \(\displaystyle 9\div9=1\)

The displayed positive values are 4, 4, 4, 4. Which explanation is valid?

Calculate before classifying

For 30, 24, 18, 12, subtraction and division give different evidence. Do not call the first quotient a common ratio until you have checked the others

1. 1Difference results All three differences are equal \(\displaystyle -6,\ -6,\ -6\)
2. 2Ratio results The ratios differ, so the displayed pattern is arithmetic only \(\displaystyle \frac45,\quad\frac34,\quad\frac23\)

A later pair can break the claim

For 6, 12, 24, 47, the first two steps multiply by 2. The last step does not: 24 multiplied by 2 would be 48. The differences 6, 12 and 23 are also unequal, so these displayed terms fit neither pattern

1. 1Do not stop early Compare the claimed operation with each displayed step, in order

A prefix is not a whole rule

The displayed values 5, 10, 20, 40 fit a geometric pattern. If no later rule is supplied, the next value could be 80 or something else. State what the given information supports

1. 1Displayed pattern All three displayed ratios are 2.
2. 2Continuation A unique later value needs an extra rule or assumption