5 mistakes learners make with linear sequences, each one named and explained.
The common difference is what multiplies n, and the constant is whatever you still need to add to land on the first term.
The 6th term of 4n - 1 is worked out with n = 5, giving 19.
The first term is where counting starts, and starting from zero is a habit worth having everywhere else.
Question. the 6th term of 4n - 1
A common answer. 19, using n = 5
The answer. 23
Why. The 6th term means n = 6, so it is 24 - 1.
The position is the n. Substitute the number you were asked for.
n squared plus 3 with n = 4 is worked out as 7 squared, or 4 times 2 plus 3.
Left to right is how everything else is read, and the priority rules are quiet.
Question. n squared + 3, when n = 4
A common answer. (4 + 3) squared, which is 49
The answer. 19
Why. Only the n is squared, so it is 16 first, and then 3 is added.
Powers happen before adding. If the brackets are not written, they are not there.
The nth term of 5, 8, 11, 14 is given as 3n + 5, using the first term, or as n + 3, using the difference as the constant.
Both numbers in the rule have to come from somewhere and the two available numbers get swapped.
Question. 5, 8, 11, 14
A common answer. 3n + 5, using the first term as the constant
The answer. 3n + 2
Why. 3n + 5 gives 8, 11, 14, which is the sequence starting one term too late.
The constant is what is left after the difference has done its work on the first term.
A sequence going up in threes is given the rule n + 3 rather than 3n plus something.
Both numbers appear in the rule and which one multiplies n is the only difference.
Question. 5, 8, 11, 14, ...
A common answer. n + 3
The answer. 3n + 2
Why. n + 3 gives 4, 5, 6, 7, which goes up in ones.
The common difference always multiplies n. Test the rule on n = 1 and n = 2.
The nth term of 5, 8, 11, 14 is given as 3n, or the 6th term of 4n - 1 is given as 24.
The difference is the part that takes real thinking, and once it is found the job feels done.
Question. 5, 8, 11, 14
A common answer. 3n
The answer. 3n + 2
Why. 3n gives 3, 6, 9, 12, which is every term two short, so 2 has to be added.
Write out what your rule actually gives and compare, term by term, with what you were given.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.