TEST 1 (MAT 370, REVIEW)

ASU ID:..................................NAME:..................................

1. Use mathematical induction to prove the following statements.

Statement 1: Given a prime number $p$ then

$$\forall \;\;n\in {\rm N}\;\; p\vert (n^p -n)$$

Statement 2: $\forall \;\;n\in {\rm N}\;\;\;6\vert (n^3 + 5n)$

2. Write binary representation for $\frac{3}{7}.$

3. Prove that the sequence

$$(1+ \frac{1}{n})^n\;\;\;n=1,\;2,\;3,\; \dots$$

is a Cauchy sequence.

4. Find the sum $S,$

$$S=1+2x+3x^2+4x^3+\dots + nx^{n-1}$$

5. Find the limit for each of the following sequences and then use the definition of limit to justify your result.

$$\frac{n^2}{3\cdot n^2 + n}$$

$$1+\frac{1}{2^{n}}+ \frac{1}{2^{2n}} +\dots +\frac{1}{2^{n^2}}$$

where $n=1,\;2,\;3,\; \dots .$