Learning objective
Sequences are functions whose domain is a subset of the integers.
Each number in a sequence is called a term. Sequences can be finite or infinite.
An arithmetic sequence is a pattern, where the same number is added to each subsequent term.
The constant is called a common difference and is usually denoted by d. The sequences will increase when d is positive or decrease when d is negative. Because the terms are increasing or decreasing by a constant amount, they will form a straight line when plotted on a graph.
Similarly, an arithmetic sequence is represented in explicit notation by the formula:
The domain of any arithmetic sequence is a subset of the integers. The domain can begin from any non-negative integer but will most often begin at 0 or 1.
For an arithmetic sequence where a term and the common difference are known we can use a similar formula:
The points on this graph represent an arithmetic sequence:
Complete the following table of values:
n | 1 | 2 | 3 | 4 |
---|---|---|---|---|
a_n |
Write the explicit rule that represents the sequence.
Find the 15th term of the sequence.
An arithmetic sequence is defined by the rule T_n=T_{n-1}+1.95 where T_1=1.5.
Write the explicit rule of this sequence.
Find T_{10}.
In an arithmetic sequence, a_7=43 and a_{14}=85.
Find the common difference.
Find the first term.
Write the explicit rule for this sequence.
Tiles were stacked in a pattern as shown:
Describe the recursive pattern and write the explicit equation for the sequence using function notation.
A table of values representing the relationship between the height of the stack and the number of tiles was partially completed.
Height of stack | 1 | 2 | 3 | 4 | 5 | 10 | 100 |
---|---|---|---|---|---|---|---|
Number of tiles | 1 | 3 |
Complete the table of values representing the relationship between the height of the stack and the number of tiles.
State the domain.
An arithmetic sequence has a common difference.
When the first term and common difference are known:
When any term and the common difference are known:
A geometric sequence is a pattern of multiplication, where the same number is multiplied to each subsequent term.
A geometric sequence is represented by the formula:
Just like arithmetic sequences, the domain of any geometric sequence is a subset of the integers, usually starting from 0 or 1.
Consider the following geometric sequence:
n | 1 | 2 | 3 | 4 |
---|---|---|---|---|
t_n | -9 | 3.6 | -1.44 | 0.576 |
Write the explicit rule for this sequence.
A geometric sequence is defined by T_n=5\cdot T_{n-1} where T_1=-1.
Write the explicit rule for this geometric sequence.
Find the 6th term.
In a geometric sequence, a_2=-\dfrac{3}{5} and a_5=\dfrac{3}{625}.
Find the common ratio.
Find the first term.
Write the explicit rule for the sequence.
A geometric sequence has a common ratio.