1a Use proof by induction to show the following for every na

1a. Use \'proof by induction\' to show the following:

\"for every natural number n, the number n2 + 5n + 6 is even.\"

1b. Use Proof by Contradiction to show:

There is no rational number x with the property that x2 = 2.

Solution

1a.

For n = 1, n2 + 5n + 6 = 1 + 5 + 6 = 12 (even)

Assuming that for some n=k, n2 + 5n + 6 is even.

That is k2 + 5k + 6 is even.

Then, for n = k+1, n2+5n+6 = (k+1)² + 5(k+1) + 6

= k2 + 1 + 2k + 5k + 5 + 6

= k2 + 5k + 6 + 6 + 2k

= (k2 + 5k + 6) + 2(k + 3)

We had assumed that k2 + 5k + 6 is even. Also, 2(k+3) is even.

Hence, n2 + 5n + 6 is also even for n = k+1

Hence, proved by induction.

1b.

x2 = 2.

Therefore, x = sqrt(2)

Let us suppose that sqrt(2) is rational.

Suppose, sqrt(2) = a/b; where a and b are whole numbers.

We additionally assume that this a/b is simplified to lowest terms, since that can obviously be done with any fraction. Notice that in order for a/b to be in simplest terms, both of a and b cannot be even. One or both must be odd. Otherwise, we could simplify a/b further.

Therefore 2 = a²/b².

Therefore a should be even.

Assuming a = 2k.

2 = (2k)²/b²

2 = 4k²/b²

2b² = 4k²

b² = 2k²

Which means that b2 is even.

And this is a contradiction. Hence, our assumption is false.

Therefore we proved that there is no rational number x with the property that x² = 2.