Quadratics

1.2 Sequences – Getting Formulas

Lesson materials located below the video overview.

Here is the constant sequence (always 1) and its leading diagonal:

Lesson 2 Sequence 1

Here are the counting numbers (given by \(n\) ) and their leading diagonal:

Lesson 2 Sequence 2

Here are the square numbers (given by \(n^{2}\)) and their leading diagonal:

Lesson 2 Sequence 3

And here are the cube numbers (given by \(n^{3}\)) and their leading diagonal:

Lesson 2 Sequence 4

And we could keep going!

Lesson 2 Sequence 5

Here’s a question:

What would be the diagonal for the sequence given by \(n^{2} + n\)?
Would it just be the sum of the two individual diagonals? (That is, sum across the rows of the diagonals.)

Lesson 2 Sequence Question

Let’s check. Here’s the sequence for \(n^{2} + n \) and its difference table.
Lesson 2 Sequence Answer

Yes … It’s leading diagonal is the sum of the diagonals \(n^{2} + n \) for and \(n\)!

 

This example suggests a strategy for finding formulas for sequences from the standard leading diagonals.

If you are given a sequence, find the leading diagonal of its difference table.

If you recognize that leading diagonal as a combination of standard leading diagonals, then you will have a candidate formula for the sequence!

This is, of course, assuming we trust patterns and choose to believe everything will work magically. But we can always check any answer we obtain from this method by plugging in numbers. So we need a third step:

Check to see if your formula actually works.

 

EXAMPLE 5: Find a formula for the sequence: 1  6  15  28  45  66  … (under the assumption we can trust patterns!)

 

Answer: Here’s its difference table:

Lesson 2 Practice 5

Is its leading diagonal a combination of any standard ones?
Lesson 2 Practice 4b

Notice that we have zeros in the fourth position onwards. This tells us we won’t be using \(n^{3}\).

Also notice we have “4” in the third position and only \(n^{2}\) has an entry in the third position, the number 2. It is clear we will need two of those twos.

We need a 5 in the second position and right now we have double 3 to give 6. This shows we need -1 of the \(n\) diagonal.
Lesson 2 Practice 5c

The “5” and the “4” are all set, and we see that we don’t need any of the 1 diagonal.

\(2n^{2} – n\) does the trick and we can check that is does indeed produce the sequence 1, 6, 15, 28, 45, 66 … . So we know we are right!

 

PRACTICE 6: Use this method to find a formula for the sequence 2, 2, 4, 8, 14, 22, 32,….

Comment: Just so you know… The answer is \(n^{2} – 3n + 4\). But see if you can get this answer from looking at the leading diagonal.

 

PRACTICE 7: Find a formula for the sequence 1 , 3, 15, 43, 93, 171, 283,….

 

PRACTICE 8: Find a formula for the sequence 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ….

 

Some Jargon:

As we have seen, a sequence with constant first differences is called linear. For example, 2, 5, 8, 11, 14, … is linear.
Lesson 2 Practice 8

A sequence with constant second differences (but not constant first differences) is called quadratic. For example, the sequence 2, 3, 6, 11, 18, 27… is quadratic. (We’ll explain this name as the course progresses.)
Lesson 2 Practice 8b

As we have seen, it seems that any quadratic sequence can be written as a formula involving \(n^{2}\) and \(n\) and 1. For example, 2, 3, 6, 11, 18, 27, … has formula \(n^{2} – 2n +3\).

A sequence with constant third differences (but not constant second differences) is called cubic. Our work suggests that any cubic sequence is given by a formula involving \(n^{3}\), \(n^{2}\), \(n\) and 1.

A sequence with constant fourth differences is called quartic. One with constant fifth differences quintic. (And constant sixth differences? Seventh differences? What might the names for these be?)

Question: What might we call a sequence that is already constant? Does it seem appropriate to call a sequence such as 3, 3, 3, 3, 3, 3, … a constant sequence?
PRACTICE 9: Let \(S(n)\) be the number of squares one can find in an \(n \times n\) grid of squares. For example, \(S(3) = 14\) because there are fourteen squares to be found in a \(3 \times 3 \) grid:
Lesson 2 Practice 9
a) Find \(S(1), S(2), S(4)\), and \(S(5)\).
b) Use the difference method (trusting patterns) to find a general formula for \(S(n)\).

OPTIONAL CHALLENGE: What’s \(1^{2} + 2^{2} + 3^{2} + 4^{2} + \cdots + 100^{2}\)?

 

Self Check: Question 1

Try the following problems to test your understanding.

If you have trouble with these, redo the practice problems in the lessons and check your answers with the solutions in the COMPANION GUIDE to this QUADRATICS course.

 

 

Self Check: Question 2

Our next question:

 

Self Check: Question 3

A third question!

 

Self Check: Question 4

Jolly ho! A fourth question!

 

Self Check: Question 5

A final question.

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