Using nth Terms of Sequences
How do I use the nth term formula for a sequence?
- nth term formulae can be given as algebraic expressions in terms of n that take positive integer values of n only
- for example,
- To find the value of the first term, substitute n = 1 into the formula
- To find the value of the second term, substitute n = 2 into the formula, and so on
- To find which term has a value of , set the formula equal to and solve the equation to find n
- for example, then solve this equation to find n = 9
- This means the 9th term has a value of
- for example, then solve this equation to find n = 9
Exam Tip
- If you are asked "which term", the question usually wants to know which value of n (e.g. n = 5, so the 5th term), not its value in the sequence
Worked example
The nth term of a sequence is given by
The first three terms are when
When :
When :
When :
Which term in the sequence is the first one to have a negative value?
We can see from part (a) that the terms are decreasing, and getting closer to zero (and then negative numbers)
Let's find when the sequence is equal to zero, and then after this, the sequence will be negative
Multiplying both sides by the denominator and solving
So the 5th term in the sequence is zero
As the sequence is decreasing, this means the 6th term will be the first negative term
But we should substitute in to check this
The 6th term