Question

Which recursive formula can be used to generate the sequence shown, where f(1) = 9.6 and n > 1? 9.6, –4.8, 2.4, –1.2, 0.6, ...
A.f(n + 1) = –0.5f(n)
B.f(n + 1) = 0.5f(n)
C.f(n + 1) = f(0.5n)
D.f(n + 1) = f(–0.5n)

Answer 1

A recursive formula needs you need to plug in the term(s) that come before the term you're trying to find to get that term you're trying to find.

In your recursive formulas, f(n) is your current term and f(n+1) refers to the next term.  Let's use answer choice A as an example (though it might not be the right answer):
You know the first term, f(1) = 9.6, making n = 1.
To find the second term, n = 2, or f(n+1) = f(1 + 1) = f(2), you are told that f(n+1) = –0.5f(n), or -0.5 * f(n). If f(1) = 9.6, where n = 1, then f(n+1) or f(2) = -0.5*9.6.
This pattern would repeat over and over again.

Now back to the problem:
The easiest way to solve this problem is by plugging in your first term and seeing what  generates the sequence.

But first, notice that the sequence alternates between negative and positive numbers. That means the recursive formula must have a negative sign in it. You can go ahead and eliminate answers B and C. 

You're left with A and D. Test out both:
Choice A, f(n + 1) = –0.5f(n)
f(1) = 9.6
f(2) = -0.5f(1) = -0.5(9.6) = -4.8
f(3) = -0.5f(2) = -0.5(-4.8) = 2.4

You don't need to go any further. Choice A works!

Choice D, f(n + 1) = f(–0.5n)
f(1) = 9.6
f(2) = f(–0.5n) = f(-0.5(1)) = f(-0.5) 

Choice D does not mathematically work.

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Answer: A

Answer 2

The recursive fromula that is used to generate the sequence 9.6, –4.8, 2.4, –1.2, 0.6, ... is  $f(n+1)=-0.5 f(n)$

Explanation :

Given f(1)=9.6, that is our first term

Sequence is 9.6, –4.8, 2.4, –1.2, 0.6, ...

There is no common difference between the sequence of numbers.

Lets find whether it has any common multiple

$\frac{-4.8}{9.6} =-\frac{1}{2} \\\\\frac{2.4}{-4.8} =-\frac{1}{2} \\\\\frac{-1.2}{2.4} =-\frac{1}{2}$

So common ratio is -1/2 or we can say -0.5

We multiply the previous term by -0.5 to get the next term

The recursive fromula that is used to generate the sequence is

$f(n+1)=-0.5 f(n)$

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