3n - 2n = 1

Question

3ⁿ - 2ⁿ = 1 

How many solutions does it have and prove them?

Answer 1

Faizan,

 

My first strategy would be to ask myself if there are any n's I can plug in to get the answer. No fancy tricks, just direct guessing. Does that work for you?

 

If that doesn't work, my next step is to attempt to graph the thing I'm unfamiliar with, in this case f(n) = 3ⁿ - 2ⁿ.

 

Now graphing it might not be straightforward. It's important to see that this is the difference between two exponential functions. Do you know what graphs of exponential functions look like? If not, try plugging in a few points to an exponential function like 3ⁿ. Ask yourself what happens when n gets very positive; what about very negative? What about in between?

 

Note that for large numbers 3ⁿ is going to be huge: 3 raised to a big power is a very big number. What about negatives? 3 raised to a very negative number is going to give you 1 divided by a very large number - would that be big or small?

 

Once you have an idea of the graph of an exponential function, think about the graph of f(n)  = 3ⁿ - 2ⁿ. When n is positive, which is bigger: 3ⁿ or 2ⁿ? What about when n is negative? What about at zero?

 

I think if you answer these questions you should get an idea of what the graph looks like. You should then be able to tell where f(n)=1... which will give you your answer!

 

Bryce