(a) Suppose you are told
\[ \begin{align}
n^2 - n + 41 &\text{ is prime} \\
\end{align} \ \]
Can you prove this using mathematical induction ?
(b) Suppoae you are given
\[ \begin{align}
1 + 3 + 5 + \ ...\ + (2n-1)\\
\end{align} \ \]
Find the sum using your knowledge of series.
Prove your result using mathematical induction.
Here is an attempted proof that
\[ \begin{align}
1 + 2 + 3 + ...\ + n &= \frac{n^2+n+1}{2}\\
\end{align} \ \]
Assume the statement is true for n = k.
Then for n = k + 1 :
\[ \begin{align}
1 + 2 + 3 + ...\ + k + k + 1
&= \frac{n^2+n+1}{2} + k + 1\\
&= ???
\end{align} \ \]
What went wrong ?