Chapter $1$

$10$ points

$1)$ Suppose your calculator only did base $10$ logarithms.

Write an expression to compute log base $2$ of $64$ using only log base $10.$

$10$ points

$2)$ Express the following summation in closed form $($an expression that can be directly computed from k$).$

$($Refer to slide $11)$

$3+5+7+9+...+2$k$+1$

$10$ points

$3)$ Proof by counterexample

Prove that the following statement is false: $2^$n $>$ n$!$ for any n $>=1$

$10$ points

$4)$ Proof by contradiction

Suppose we know the shortest path from A to D is A to B to C to D$.$ Using the technique of proof by contradiction, show why the shortest path from A to C must be A to B to C$.$

$10$ points

$5)$ Induction proofs.

a$.$ Prove by induction:

n

sum $(2$i$-1)=$ n$^2$

i$=1$

Note: sum is intended to be the summation symbol, and $^$ means what follows is an exponent

$10$ points

$6)$ Given: T$(1)=3$

T$($N$)=$ T$($N$-1)+3,$ N$>1$

What would the value of T$(10)$ be$?$

$10$ points

$7)$ For the problem above, is there a formula that could directly calculate T$($N$)?$

$10$ points

$8)$ Using induction, prove that your formula for the previous problem is correct.

$20$ points

$9)$ Write a generic class $($using Java generics$)$ called Pair that stores two values of the generic type. It should have a constructor to set them, two methods getfirst$()$ and getsecond$()$ to retrieve them, and a method called swap$()$ that exchanges them. Include a "main" method that creates a Pair of strings and swaps them, printing the before and after values. You should provide a source file$($s$)$ as your solution to this problem.