For each of these statements about natural numbers, decide if it's true or false and prove your claim using

Question

only the properties of natural numbers.
a)If nЄN and n^2 + (n+1)^2 = (n+2)^2, then n=3.
b)For all nЄN, it is false that (n-1)^3 + n^3 = (n+1)^3

Answer 1

Just expand them and solve.

a) n²+(n+1)²=(n+2)²
n²+(n²+2n+1)=n²+4n+4
n²-2n-3=0
(n+1)(n-3)=0
n=-1 or n=3

But since n is a natural number, n=-1 is rejected, leaving you with n=3.

b) (n-1)³+n³=(n+1)³
n³-3n²+3n-1+n³=n³+3n²+3n+1
n³-6n²-2=0

Which has no natural number solutions. Therefore, the statement is true.

Answer 2

a) Asssume n^2 + (n + 1)^2 = (n + 2)^2

Expanding , we get

n^2 + n^2 + 2n + 1 = n^2 + 4n + 4

Simplifying,

2n^2 + 2n + 1 = n^2 + 4n + 4

Moving everything over to the left hand side,

n^2 - 2n - 3 = 0

Factoring this, we get
(n - 3)(n + 1) = 0

It follows that n - 3 = 0, or n + 1 = 0, meaning
n = 3 or n = -1
However, since n is an element of the natural numbers, we reject the solution n = -1, and n = 3.
The statement is true.

b) We want to determine whether this statement is false:
(n - 1)^3 + n^3 = (n + 1)^3

Factoring the left hand side as a sum of cubes, we get

[(n - 1) + n] [(n - 1)^2 - n(n - 1) + n^2 ] = (n + 1)^3
[2n - 1] [(n - 1) (n - 1 - n) + n^2] = (n + 1)^3
[2n - 1] [(n - 1)(-1) + n^2] = (n + 1)^3
(2n - 1)[-n + 1 + n^2] = (n + 1)^3
(2n - 1)(n^2 - n + 1) = (n + 1)^3

Expanding both sides,

2n^3 - 2n^2 + 2n - n^2 + n - 1 = n^3 + 3n^2 + 3n + 1

Simplifying,

2n^3 - 3n^2 + 3n - 1 = n^3 + 3n^2 + 3n + 1

Moving everything over to the left hand side,

n^3 - 2 = 0

n^3 = 2

This has no solutions which are natural numbers. Therefore, this statement is true.

Answer 3

Both these statements are true.
Just expand by the binomial theorem and simplify.
The first equation reduces to n²-2n-3=0,
i.e., n²-2n = 3 or n(n-2) = 3
This says that n divides 3,so n = 1 or 3,
because n is a natural number and 3 is prime.
But only 3 satisfies the equation.
Next, the second equation reduces to
n³-6n²-2=0 or n²(n-6) =2.
That says that n² divides 2. So n = 1 because 2 is
a prime number.
But 1 doesn't satisfy the equation.
Hope that helps!!