Sequences & Series · Ordinary Level
Patterns & the Term Rule
Ordinary Level · Class 1 · Tap NEXT to begin
Section 1 of 2
Spotting the Pattern
A sequence obeys a rule
A sequence is a list of numbers that follows a rule. Each number is a term: the first is $T_1$, the second $T_2$, and so on.
Worked example — find the next number
Find the next number in each: (i) $2, 4, 6, \_$; (ii) $5, 8, 11, \_$; (iii) $7, 13, 19, \_$.
(i) going up in $2$: next is $8$
(ii) going up in $3$: next is $14$
(iii) going up in $6$: next is $25$
$8,\ 14,\ 25$
You try
Find the next number: (i) $28, 24, 20, \_$; (ii) $25, 19, 13, \_$.
They are going down.
(i) down in $4$: next is $16$
(ii) down in $6$: next is $7$
$16,\ 7$
Section 2 of 2
The Term Rule, Tₙ
Substitute n
A rule like $T_n$ gives any term. To find a term, put its position in for $n$. So $T_1$ uses $n = 1$, $T_{10}$ uses $n = 10$.
Worked example — using the rule
$T_n = 3n + 2$. Find $T_1$, $T_2$, $T_3$.
$T_1 = 3(1) + 2 = 5$
$T_2 = 3(2) + 2 = 8$
$T_3 = 3(3) + 2 = 11$
$5,\ 8,\ 11$
Worked example — a bigger term
$T_n = 5n - 1$. Find $T_1$ and $T_{10}$.
$T_1 = 5(1) - 1 = 4$
$T_{10} = 5(10) - 1 = 49$
$T_1 = 4,\ T_{10} = 49$
Worked example — with a general term
$T_n = 5 - 7n$. Find $T_3$, $T_5$ and $T_{n+1}$.
$T_3 = 5 - 7(3) = -16$; $T_5 = 5 - 7(5) = -30$
$T_{n+1} = 5 - 7(n+1) = 5 - 7n - 7 = -7n - 2$
$-16,\ -30,\ -7n-2$
You try
$T_n = 7n - 1$. Find the first $3$ terms.
Put $n = 1, 2, 3$.
$T_1 = 6,\ T_2 = 13,\ T_3 = 20$
$6,\ 13,\ 20$
You try
$T_n = 1 - 5n$. Find the first $3$ terms.
Put $n = 1, 2, 3$.
$T_1 = 1 - 5 = -4$; $T_2 = 1 - 10 = -9$; $T_3 = 1 - 15 = -14$
$-4,\ -9,\ -14$
That’s Class 1.
Spotting a pattern and using the term rule $T_n$. Class 2: arithmetic sequences.