   SEARCH HOME Math Central Quandaries & Queries  Question from Rita, a student: I am having trouble understanding how to write formulas (I should say create formulas) from a given sequence. It does not matter what sort of sequence it is. The confusion remains. I have not found a textbook or easy math book that explains this process for the average Joe to grasp. Here are the two questions: Write a recursive formula for the sequence 9, -18, 36, -72, ... I understand that a_1 = first term given, which is 9 in this case. I also found that -2 is multiplied by each term before to find the term that follows. But how do I create the rest of the formula? Write a recursive formula for the sequence 3, 3(sqrt{3}), 9, 9(sqrt{3}) Again, the first term is 3, which is written in the book as a_1. How do I create the needed formula? Hi Rita,

The subscript on the letter a tells you which term it is so

the first term is a1
the second term is a2
the third term is a3
and so on

When we want to write an expression that is valid for a group of terms we often use a letter for the subscript and then describe the values that the letter can take. So for example you might say an = 3n for n = 1, 2, 3 or 4 rather than write a1 = 3, a2 = 6, a3 = 9 and a4 = 12.

In your first example you correctly say that a1 = 9 and you also correctly say that -2 is multiplied by each term before to find the term that follows. Thus

the second term a2 is -2 × a1

the third term a3 is -2 × a2

the fourth term a4 is -2 × a3

and so on.

So how do you write a general expression for all the terms beyond the first? To use mathematical notation in place of your phrase -2 is multiplied by each term before to find the term that follows I would say that

the term numbered n + 1 is -2 × an or

an+1 = -2 × an

So here is my answer

A recursive formula for the sequence 9, -18, 36, -72, ...is

a1 = 9 and an+1 = -2 × an for all n > 1.

You try the second example and write back if you need further assistance.

Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.