Instructions: 3 hours, 9 problems counted equally; show your work and justify your answers.

1. Let a be a real number other than 1. Use induction to show that for each positive integer n,

$$
\sum _ { k = 0 } ^ { n - 1 } a ^ { k } = ( a ^ { n } - 1 ) / ( a - 1 ) .
$$

2. Let $f : \mathbb { R } \to \mathbb { R }$ be a function and let a be a real number.
   Let S be the statement $\operatorname { \arg f } f$ has a local minimum at $^ { a , }$ then $f ^ { \prime } ( a ) = 0 \ \mathrm { o r } \ f$ is not differentiable at $\pmb { a } ^ { \dag }$

a) Write the contrapositive of S.

)Write the converse of S.

c) Which of the statements in (a) and (b) are true?
If either statement is false, give a counterexample.

3. Let a be the complex number $- 2 + 2 { \sqrt { 3 } } i$ . Express $\alpha ^ { 3 }$ and the two square roots of α in the standard form $a + b i$

4.Prove that there is no $n \in \mathbb { N }$ so that $( 2 1 n - 3 ) / 4$ and $( 1 5 n + 2 ) / 4$ are both integers.

5. Let $T : V \to W$ be an injective linear transformation and suppose $v _ { 1 } , \ldots , v _ { n }$ are linearly independent vectors in V. Prove that the vectors $T ( v _ { 1 } ) , \ldots , T ( v _ { n } )$ are linearly independent in W.

6. Find all real numbers x for which the series $\sum _ { n = 1 } ^ { \infty } { \frac { x ^ { n } } { \sqrt { n ^ { 2 } + 1 } } }$ converges.
   [Be sure to justify your analysis.]

7. Suppose $f , g : \mathbb { R } \to \mathbb { R }$ and $a \in \mathbb { R }$

State the $\epsilon - \delta$ definition of what it means for f to be continuous at a.

b) Prove that if f and g are continuous at 3, then their sum $f + g$ is also continuous at 3.

8. Evaluate the line integral $\oint _ { C } P d x + Q d y$ where $P = y ^ { 2 } , Q = 2 x - 3 y$ ,and C is the circle $x ^ { 2 } + y ^ { 2 } = 9$

9. Suppose $f : \mathbb { R } \to \mathbb { R } ^ { 2 }$ and $g : \mathbb { R } ^ { 2 }  \mathbb { R }$ are differentiable.
   Prove that the composite function $f \circ g$ cannot have a differentiable inverse.