A recursive formula for 9, -18, 36, -27, ...

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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 a_n = 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 × a_n or

a_n+1 = -2 × a_n

So here is my answer

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

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

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

Penny

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