Algebra

UK Secondary (7-11)

Rules for describing sequences

Lesson

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:

We can use a table to more easily understand the 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`=5`x`.

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.

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`×2−4B$A=N^2-4$

`A`=`N`2−4C$A=2+N-4$

`A`=2+`N`−4D$A=2\times\left(N-4\right)$

`A`=2×(`N`−4)A$A=N\times2-4$

`A`=`N`×2−4B$A=N^2-4$

`A`=`N`2−4C$A=2+N-4$

`A`=2+`N`−4D

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

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?