SOLUTION: Trying to help my friend with his algebra, and I'm stumped as to what they're going for on this one: Explain why: {{{(sqrt(n)-sqrt(n+1))(sqrt(n)-sqrt(n-1))=1}}} is true for {{{n

Question 226228: Trying to help my friend with his algebra, and I'm stumped as to what they're going for on this one:
Explain why:

is true for two different ways.
a) Provide two numerical values.
b) Show that is it true in general.

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
The equation should be . Make sure that you have the correct equation.

a) Let's show that is true for the values and

Let's show the equation is true for n=1:

Start with the given equation.

Plug in

Combine like terms.

Take the square root of 1 to get 1. Take the square root of 0 to get 0.

Combine like terms.

Multiply.

So is true when

-----------------------

Let's show the equation is true for n=2:

Start with the given equation.

Plug in

Subtract

Take the square root of 1 to get 1.

FOIL

Multiply

Combine like terms.

So is true when

====================================================================

b)

Now let's show that is true for any value of 'n' such that

Start with the given equation.

Let and

Replace each with 'x' and replace each with 'y'

FOIL

Combine like terms.

Plug in and

Square each square root to eliminate them (note: because , this means that which means that each radicand is positive. So we can avoid absolute values here).

Distribute

Combine like terms.

So this shows that is true for any value of 'n' such that
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