Question

Rakesh and Tessa were asked to find an explicit formula for the sequence 100, 50, 25,12.5. Rakesh said the formula is f(n)=100*(1/2)^n-1, and tessa said the formula is f(n)=200*(1/2)^n. Which one of them is right?

Answer 1

Both

Explanation:

In a geometric sequence, the ratio between successive terms is constant. This means that we can move from any term to the next one by multiplying by a constant value. Let's calculate this ratio over the first few terms:

12.5/25 = 25/50 ​ = 50/100 = 1/2

We see that the constant ratio between successive terms is 1/2

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Both Rakesh and Tessa got a correct explicit formula.

I did this khan academy and it takes forever to type all of the steps but its both of them.

Answer 2

200*(1/2)^n

Explanation:

This question can be solved by substituting the value of n= 1,2,3,4 in the both of the expression as there are four terms in the series.

For first expression 100*(1/2)^n-1

`Term1 = 100*(1/2)^1 - 1 = 49\\\\Term2 = 100*(1/2)^2 - 1 = 24\\\\Term3 = 100*(1/2)^3 - 1 = 11.5\\\\Term4 = 100*(1/2)^16 - 1 = 5.25\\\\Hence \ the \ series \ is \ 49, 24, 11.5, 5.25\\\\For \ expression 200*(1/2)^n\\Term1 = 200*(1/2)^1 = 100\\\\Term2 = 200*(1/2)^2 = 50\\\\Term3 = 200*(1/2)^3 = 25\\\\Term4 = 200*(1/2)^2 = 12.5\\\\Hence \ the \ series \ is \ 100, 50, 25,12.5\\`

Thus formula 200*(1/2)^n correctly expresses the sequence 100, 50, 25,12.5