Sequences game: arithmetic and geometric progressions
A sequences game where you type, draw and build. The next term, term 50 or the sum of the first terms are typed in, with no options to pick from. In the dot figures you draw figure 4 by tapping squares, seeing which dots stay fixed and which are added at each step. And the general term is built with − and +: dn + c for arithmetic and a × rⁿ⁻¹ for geometric sequences, while the game shows the first four terms your formula gives next to those of the sequence. There is also sorting sequences into arithmetic, geometric or neither.
Mathematics · Secondary, Sixth form
From the first term to the hundredth there are 99 steps, not 100. That slip is in almost every exam, and here it comes up as an option again and again until it sticks: aₙ = a₁ + (n − 1)d and aₙ = a₁ × rⁿ⁻¹.
In a dot pattern that grows by 3 with 2 fixed dots, the general term 3n + 2 is not a formula out of nowhere: it is the 2 that never change plus 3 for each figure. Colouring them when you answer turns the algebra into something you can see.
Telling arithmetic from geometric is the first step in every exercise. The wrong options include adding the ratio instead of multiplying or carrying on adding the last difference, and there are also sequences that are neither, like the squares or Fibonacci.
"The sequence is 5, 8, 11, 14… and the general term has to be built. The student sets d = 3 and c = 5, and the formula gives 8, 11, 14, 17: one step too many. They lower c to 2 and the four terms turn green: aₙ = 3n + 2."
Prefer to learn without screens? Try this analogue version:
With matchsticks or toothpicks, make a row of joined squares: one, two and three squares.
Count the sticks in each figure (4, 7, 10…) and write down the sequence.
Find what stays the same and what is added with each square, and write the general term.
Use your formula to predict how many sticks 50 squares need, and check it with figures 4 and 5.
If you like Next in Line, these activities work similar skills:
Another maths game: the powers of 10 are a geometric sequence with ratio 10.
Solve first-degree equations, which is what you need to find which term of a sequence equals a given number.
Count the spirals on a pine cone or a sunflower head: they are almost always Fibonacci numbers, a famous sequence that is neither arithmetic nor geometric.
More practice on · Maths
Sequences and Progressions
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