Are these three expressions equivalent?

Question

I was doing a test problem at Khan Academy in the geometric sequence section. The problem gave an iterative function and I'm supposed to write the explicit function:

I kept on trying:

$$2.2 \cdot -5^{n-1} $$ $$2.2 \cdot -5^{(n-1)} $$ Both failed, I was tearing my hair out until I did: $$2.2 \cdot (-5)^{n-1} $$

And it worked. Is this most likely a bug or am I confused about order of operations and fail to see that -5 needs to be in brackets? All three answers are equivalent, no?

Answer 1

No, they are not equivalent.

$$2.2 \cdot -5^{n-1} = 2.2 \cdot -1 \cdot 5^{n-1}$$

The former is evaluated as $5$ to the $n-1$ power multiplied by $-1$ while the latter is $-5$ to the $n-1$ power. Remember: exponentiation is performed before multiplication.

They will have different signs when $n$ is odd:

$$-5^{3-1} = -1 \cdot 5^2 = -1 \cdot 5 \cdot 5 = -25$$ $$(-5)^{3-1} = (-5)^2 = -5 \cdot -5 = 25$$

Answer 2

$$g(1)(-5)^{n-1}=g(1)(-5)^{(n-1)} \neq g(1).-5^{n-1}$$ Not the same answer when n is odd but $$g(n)=g(1)(-5)^{n-1}$$

Is the correct formula...

$$g(2)=-5g(1)$$ $$g(3)=5^2g(1)$$ $$g(4)=-5^3g(1)$$ The sign alternate between + and -