Math CentralQuandaries & Queries


Question from zoe, a student:

what is the nth term for the following sequence...
5, 2, -1, -4

Hi Zoe,

If you look at the terms in your sequence you will see that the terms all decrease by 3 each time. This constant difference makes your sequence an arithmetic sequence. There is a formula for finding the nth term of an arithmetic sequence:

tn = a + (n-1)d where tn represents the nth term
a represents the first term
n represents the number of terms
d represents the common difference between the terms

In your sequence, a = 5, and d = -3. Since we are looking for an expression for the nth term, we leave n as n and we get that:

tn = 5 + (n-1)(-3)

If you wanted to expand and simplify this expression just distribute the -3 through the brackets and combine the constants:

tn = 5 - 3n +3
tn = 8 - 3n

You can verify that this formula does in fact produce the terms you already know:

Your first term (when n=1) t1 = 8 - 3(1) = 5
Your second term (n = 2) t2 = 8 - 3(2) = 2
Your third term (n = 3) t3 = 8 - 3(3) = -1
Your fourth term (n = 4) t4 = 8 - 3(4) = -4

And now that you have the formula, you could find the value of any term, you'd just need to know what n value you wanted. Suppose you wanted to find the 84th term. All you'd have to do is substitute 84 for n and evaluate the right-hand side:

t84 = 8 - 3(84) = 8 - 252 = -244

Hope this helps,

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