Why is $0.5^x + 0.5^x = 0.5^{x-1} $

Question

I believe that the following is always true:

$0.5^x + 0.5^x = 0.5^{x-1} $

but I do not know why. I've tried to prove it but am unsure how. Is it always true? And is there a proof?

Answer 1

You may think as follows

$$ 0.5^x+0.5^x=\Big(\frac{1}{2}\Big)^x+\Big(\frac{1}{2}\Big)^x=2\cdot \Big(\frac{1}{2}\Big)^x = 2\cdot \frac{1^x}{2^x}= \frac{2\cdot 1^x}{2^x}=\frac{2\cdot 1}{2\cdot 2^{x-1}}=\frac{1}{2^{x-1}}=\Big(\frac{1}{2}\Big)^{x-1}. $$

Hope this helped.

Answer 2

Hint:

$$0.5^x+0.5^x=2·0.5^x=2·\bigg(\frac{1}{2}\bigg)^x=\bigg(\frac{1}{2}\bigg)^{x-1}$$

Answer 3

$$0.5^x + 0.5^x=2\times(0.5)^x=2 ×\left(\frac12\right)^x=2×\left(\frac1{2^x}\right)=\left(\frac12\right)^{x-1}=0.5^{x-1}$$

Answer 4

We have $$\left(\frac{1}{2}\right)^x+\left(\frac{1}{2}\right)^x=2\cdot \left(\frac{1}{2}\right)^x= \frac{2}{2^x}=2^{1-x}$$

Answer 5

From right to left:

$0.5^{x-1}= \dfrac{0.5^x}{0.5^1}= \dfrac{2(0.5^x)}{2 (0.5)}=$

$\dfrac{2(0.5^x)}{1}= 2(0.5^x)= 0.5^x +0.5^x.$