Rules for Describing Sequences
We have already looked at how to use algebra to describe number patterns, but we can also describe patterns in shapes and geometry in a similar way.
Let's have a look at a common pattern:
| Flowers ($x$x) | $1$1 | $2$2 | $3$3 | $4$4 |
| Petals ($y$y) | $5$5 | $10$10 | $15$15 | $20$20 |
In this pattern, $x$x represents the step we are at (that is, the number of flowers) and $y$y represents the total number of petals at that step.
Notice that $y$y is increasing by $5$5 each time - in particular, the value of $y$y is always equal to $5$5 times the value of $x$x. We can express this as the algebraic rule $y=5x$y=5x.
Now that we have this rule, we can use it to predict future results. For example, if we wanted to know the total number of petals when there were $10$10 flowers, we have that $x=10$x=10 and so $y=5\times10=50$y=5×10=50 petals.
Worked Examples
Question 1
Use the rule to complete the table of values:
"The starting number is doubled, then $4$4 is subtracted."
Starting Number ($N$N) $12$12 $13$13 $14$14 $15$15 Answer ($A$A) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Which of these equations describes the rule above?
$A=2\times\left(N-4\right)$A=2×(N−4)
A$A=N\times2-4$A=N×2−4
B$A=N^2-4$A=N2−4
C$A=2+N-4$A=2+N−4
D
Question 2
a) Complete the table of values for the pattern shown
b) Write an algebraic rule for the relationship
c) Plot the graph of the line on the plane
Question 3
Matches were used to create the pattern pictured,
a) Complete the table of values
b) Write a formula that describes the relationship between the number of matches, $m$m, and the number of triangles, $t$t.
c) How many matches are required to make $53$53 triangles using this pattern?