A geometric sequence g starts 80, 40, . . . Explain how to use the recursive definition to find g(100). (Don't actually determine the value.)

Answer 1

In order to use any recursive formula you need the previous term (a_n-1 or g(n-1)). Meaning, if you want to find the 100th term which in this case is represented as g(100) you will also need the 99th term or g(99). However, because you only have g(1) = 80 and g(2) = 40, in order to find g(100) you will also need all the terms in between what you have, g(2) and g(1), and what you need, g(100). As a result, it is much faster to use the explicit formula to determine the value of g(100) since the recursive formula is much more tedious.

General formula for recursive: a_n = a_n-1*r

General formula for recursive with the variables given: g(n) = g(n-1)*r

So, g(100) = g(99)*(0.5), but using recursive we would need g(99) so g(99) = g(98)*(0.5), g(98) = g(97)*(0.5) ... g(3) = g(2)*(0.5), g(2) = g(1)*(0.5)

Answer 2

To rephrase this, there is a geometric sequence called g. g(1) is the first value in the sequence, g(2) the second... g(100) is the hundredth.

We are given: g(1) = 80

g(2) = 40

This looks like we have a pattern starting with 80, and each subsequent value in the sequence is the previous one divided by 2 (or times one half).

If this is the case, then we can write a formula for the sequence as:

g(n) = 80 * (1/2)^(n-1)

g(n) = 80 times one half raised to the power of n - 1 [placement in the sequence - 1]

Therefore, g(100) = 80 * (1/2)^99