Arithmetic Sequences and Simultaneous Equations.
Investigation 1
Arithmetic sequences have a common
difference between consecutive terms.
For example
7, 10, 13, 16, 19, 22, …
-8, -3, 2, 7, 12, 17, …
17, 9, 1, -7, -15, -23, . .
Write down six consecutive terms of an
arithmetic sequence.
Use these terms to form a pair of
simultaneous equations by using them in
order to fill the gaps in the equations
shown below.
Investigation 3
Repeat Investigation 1 but this time use
12 consecutive terms from an arithmetic
sequence to create a set of three
simultaneous equations in 3 unknowns.
For example: using the terms 3, 5, 7, 9, 11,
13, 15, 17, 19, 21, 23, 25, … we get the
equations
3x + 5y + 7z = 9
11x + 13y + 15z = 17
19x + 21y + 23z = 25
Solve the set of simultaneous equations
that you created from your sequence.
__ x + __ y = __
___ x + __y = __
For example: using the sequence
-3, 1, 5, 9, 13, 17, you would form the
equations
-3x + y = 5
9x + 13y = 17
Solve the pair of equations that you have
created.
Try another pair of equations.
What did you discover?
Why?
Try another example.
What do you notice?
Investigation 4.
Repeat Investigation 3, but this time sue
terms from different arithmetic sequences
to form each of the equations.
Investigation 5
Repeat Investigation 1, but this time use
consecutive terms of the Fibonacci
sequence.
Investigation 6 (Ideally done at a
different time to Investigations 1 to 5)
Investigation 2
Repeat Investigation 1, but this time use
three consecutive terms from one
arithmetic sequence to create one of the
equations, and three consecutive terms
from a different sequence to create the
other equation.
Write the equations of three different lines
that pass through (-1, 2).
Write these equations in the form
ax + by + cz = d
What did you discover?
The lines x = -1 and y = 2 pass through
(-1, 2). Does your observation include
those lines?
Adapted from RISP- Rich starting points
by Jonny Griffiths on page 23 in
Mathematics Teaching 212.
What do you notice about the
coefficients?
Roger Harvey, Victoria University July
2009