6 mistakes learners make with quadratic and other sequences, each one named and explained.
When the difference between terms is not constant, look at the differences of the differences, or at what is being multiplied.
100 is said to be a term of 7n - 2, at about the 14th or 15th position.
The arithmetic was done properly and the answer came out close to a whole number, so it looked near enough.
Question. is 100 a term of 7n - 2
A common answer. yes, at about the 14th term
The answer. no
Why. n would have to be 102 divided by 7, which is not a whole number, and there is no fourteen and a half th term.
A position has to be a whole number. Solve it, then check the answer is one.
100 is ruled out of the sequence 7n - 2 because it is even.
Odd and even is a real pattern that often does decide these questions, and it is much quicker than solving.
Question. is 100 a term of 7n - 2
A common answer. no, because 100 is even
The answer. no, because 102 divided by 7 is not a whole number
Why. The sequence has both odd and even terms, 5, 12, 19, 26, so being even rules nothing out.
Solve it. A property you noticed is a hunch until the equation agrees.
3, 8, 15, 24 is continued as 33 and 42, using a difference of 9.
The last gap you looked at was 9, and using it again is the natural move if the gaps are never compared.
Question. 3, 8, 15, 24
A common answer. 33 and 42, adding 9 each time
The answer. 35 and 48
Why. The gaps are 5, 7, 9, which grow by 2 each time, so the next gaps are 11 and 13.
When the differences change, look at the differences of the differences. Constant there means a quadratic rule.
A quadratic sequence is continued with a second difference that does not match the earlier terms.
Two layers of differences is one more layer than most sequences need.
Question. 3, 8, 15, 24, ...
A common answer. 34 and 45
The answer. 35 and 48
Why. The first differences are 5, 7, 9, so they continue 11 and 13.
Write both rows of differences under the sequence and extend the bottom row first.
A term is found by adding two terms that are not the two immediately before it.
Keeping track of which pair is current takes attention that the addition itself has already used.
Question. 2, 5, 7, 12, ... each term is the sum of the two before
A common answer. 17, from 5 + 12
The answer. 19, from 7 + 12
Why. The two terms before the next one are 7 and 12.
Point at the last two terms with two fingers before adding. Then move both along together.
The nth term of 2, 4, 8, 16 is given as 2n or n squared.
Almost every sequence met so far has gone up by adding, so the first thing looked for is a common difference.
Question. 2, 4, 8, 16
A common answer. 2n, which gives 2, 4, 6, 8
The answer. 2 to the power n
Why. Each term is double the one before, so the rule multiplies rather than adds.
Check the differences. If they are not constant, ask what is being multiplied instead.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.