53 apples since 22 plus 31 = 53.
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
6 subtract
the main number to get youre answer and 6 is what i got
1) first pick red 5/9 blue 4/9
2) second pick when red has been picked red now becomes 4/8, blue stays 4/8
2) Second pick when blue has been picked blue now becomes 3/8, red stays 5/8
Mind you, you only need both different colours so:
first pick red 5/9 x sec pick blue 4/8 = 20/72
and first pick blue 4/9 x sec pick red 5/8 = 20/72
add both up 20/72 + 20/72 = 40/72 = 5/9