Quadratics

1.1 Sequences… For When You Believe in Patterns

Lesson materials located below the video overview.

If you do indeed believe that patterns hold true … what would you say is the next number in this sequence of numbers?

Series 1

No doubt you noticed the constant difference of three between terms and so would guess the next entry to be 20. Of course, there is no reason to believe that all differences will forever remain constant, but noticing this for now makes “20” an intelligent guess.

Some people describe a sequence as “linear” if the difference from term to term is constant. In our example we have constant first differences and so it is linear.

Series 2

If we imagine our sequence as set of data values:

Sequence table

and plot them on a graph, then we see a straight line picture – explaining the name linear:

Plot

(At step one place to the right matches a rise of three units.)

Of course, not all sequences are linear: In this sequence, for instance:

Sequence 3

The “second differences,” not the first, turn out to be constant:

Sequence 4

 

YOUR TURN…

All solutions to the practice problems that follow appear in the COMPANION GUIDE to this QUADRATICS course.

 

PRACTICE 1: Make a prediction for the term after 38 in the sequence above.

 

PRACTICE 2:
(a) 
Show that for the following sequence the third differences are constant.

0  2  20  72  176  350  612 …

(b) How many differences must one complete in the sequence below, the powers of two, to first see row of constant differences?

1  2  4  8  16  32  64  128  256 …

PRACTICE 3:
(a) 
The square numbers begin: 1, 4, 9, 16, 25, 36, 49, 64, 81 … (The \(n\)th square number is \(n^{2}\).)  Is there a row of their difference table that is constant?

(b) The cube numbers begin: 1, 8, 27. 64, 125, 216, 343, 512, 729, … (The \(n\)th cube number is \(n^{3}\).) Is there a row of their difference table that is constant?

 

Here is a sequence and a table of all its differences. We’ve circled its leading diagonal.

Sequence diagonal 1

Now here’s just a leading diagonal. Check that you can now construct the original sequence from it.

Sequence diagonal 2

 

PRACTICE 4: Do it! Show that you can fill in all the blanks of the entire table just from knowing the leading diagonal.

 

Thus:

TO KNOW THE LEADING DIAGONAL OF A DIFFERENCE TABLE IS TO KNOW THE SEQUENCE!

 

So let’s get to know some leading diagonals! We’ll make that the start of the next lesson.

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