A sequence is a list of things placed in a certain order. This lesson will teach you how to solve questions relating to sequences.
What is an arithmetic sequence
When you come across an arithmetic sequence, the difference between one term and the next one is constant. So for example, when we have a sequence of 1,2,3,4,5, this is arithmetic since each number is simply +1 to the previous number. There’s actually a special name for the +1 in this situation…
What is common difference
What is the common difference definition? When you figure out the constant difference between terms in an arithmetic sequence, you’ve found the common difference! In the previous example, 1 is the common difference between the terms.
How to find the common difference
When you’ve got an arithmetic sequence, how do you find the common difference? Firstly, your sequence should be in order. If you know that the sequence is arithmetic, then simply find the difference between the 2nd and 1st term and you’ll find the common difference. If you wanted to check your answer, feel free to find the difference between any consecutive terms and you should find that you’ll get the same answer for your common difference.
On the other hand, if you’re not sure if a sequence is arithmetic, but you find that there’s a constant difference between all the terms, you’ve proven that the sequence is arithmetic.
How to find the nth term
When you are trying to find the nth term of an arithmetic sequence, use the following arithmetic sequence formula:
tn is the nth term, and t1 is the first term. d is the common difference in the arithmetic sequence. So if we use the example from above where we have a sequence of 1,2,3,4,5, if we had to find what would be the 6th term to come after the number 5 in our sequence, we’d get:
This makes sense since we know the next number to come should be 6. Use the above formula and it will help you in finding the nth term. Let’s look at more examples on arithmetic sequences.
Arithmetic sequence formula
Consider the arithmetic sequence: 5, 9, 13, 17, …
a) Identify the common difference.
Common difference is the difference between successive terms. We can pick any pair of successive terms to calculate the common difference. In this question, we have four terms:
We can pick any two successive terms from here. So:
We now know the common difference of this arithmetic sequence is 4
b) Determine the seventh term of the sequence.
We can use this equation to find the answer:
d: common difference
We are looking for the seventh term. So:
We can verify the answer by finding the terms up to the seventh term one by one. To do that, we just need to add the common difference to the last term:
An arithmetic sequence (arithmetic progression) is a number sequence with a common difference between successive terms. By using the arithmetic sequence formula, we can easily find the value of a term and the common difference in the sequence.
• arithmetic sequence: a sequence with a common difference between successive terms
• The nth term, tn ,of an arithmetic sequence: tn=t1+(n−1)d
where, tn: nth term
t1: first term
d : common difference
Arithmetic sequence formula
Consider the arithmetic sequence: 5, 9, 13, 17, … .
Identify the common difference.
Determine the seventh term of the sequence.
Which term in the sequence has a value of 85?
Determine t1,d,tn for the sequences in which two terms are given
Three consecutive terms of an arithmetic sequence are written in the form: 1+2x,7x,3+4x
Solve for the value of x.