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
The correct answer is $1820.
The formula for continuously compounded interest is
A = Pe^(rt), where P is the amount of principal, r is the interest rate expressed as a decimal number, and t is the number of years. Using our information, we have:
A = 950*e^(0.065*10) = 1819.76 ≈ 1820
Easiest ways to solve them are to take the denominator and numerator and and divide the one you are asked to use to answer and divide and see if it works on both.
Example: first 105 / 15 = 7 ) Then: 105 / 7 = 15
119 ) 119 / 7 = 17
Hope I helped. :D
