Base case: if <em>n</em> = 1, then
1² - 1 = 0
which is even.
Induction hypothesis: assume the statement is true for <em>n</em> = <em>k</em>, namely that <em>k</em> ² - <em>k</em> is even. This means that <em>k</em> ² - <em>k</em> = 2<em>m</em> for some integer <em>m</em>.
Induction step: show that the assumption implies (<em>k</em> + 1)² - (<em>k</em> + 1) is also even. We have
(<em>k</em> + 1)² - (<em>k</em> + 1) = <em>k</em> ² + 2<em>k</em> + 1 - <em>k</em> - 1
… = (<em>k</em> ² - <em>k</em>) + 2<em>k</em>
… = 2<em>m</em> + 2<em>k</em>
… = 2 (<em>m</em> + <em>k</em>)
which is clearly even. QED
Answer:
Step-by-step explanation:
All side lengths of a square are equal. The three squares create a right triangle, and the hypotenuse of this triangle represents the side length of the square.
All right triangles must follow the Pythagorean Theorem 
where 
is the hypotenuse of the triangle.
Therefore, let 
be the side length of this largest square. Since the area of a square with side length 
is given by 
, 
will represent the area of the square:
Scale factor is defined as a number which multiplies or scales some quantity.
The shape of the figure remains same, the image becomes larger or smaller.
If the scale factor is greater than one, then the image is enlarged.
If the scale factor is less than 1 but greater that 0, then the image becomes smaller.
In the above image, AD = 2 units.
It's corresponding side PS = 3 units
Similarly AB = 1 unit and PQ = 1.5 units
Hence , 1 * 1.5 = 1.5
2*1.5 = 3
Hence, the scale factor of the given image is 1.5 ... Answer
The square numbers of 45?
I don't really know what you mean by that but if you're looking for the square numbers before 45, they are as follows:
1
2
4
9
16
25
36
Hope this helps :)
