Quiz 5

Justify all answers with neat and organized work. Clearly indicate your answers. 20 points possible.

1. (4 pts.) The formula

1+2+3+...+n= n(n+1) 2

is true for all integers n ³ 1. Use this fact to solve both of these problems.

(a) If t is an integer and t ³ 4, find a formula for 1+2+3+...+(t-3).

(b) If n is an integer and n ³ 1, find a formula for 4+8+12+...+4n+8.

2. (3 pts.) For any nonnegative number c, define p_c, the power function with exponent c, as follows:

pc(x)=xc      for each nonnegative real number x.

Draw the graphs of the power functions p₄ and p₅ on the same set of axes. When 0 < x < 1, which is greater: x⁴ or x⁵? When x > 1, which is greater: x⁴ or x⁵?

3. (5 pts.) Suppose a sequence satisfies the given recurrence relation and initial conditions. Find an explicit formula for the sequence.

an=5an-1-6an-2, for all integers n ³ 2

a0=1,    a1=0

4. (4 pts.) Prove the following statement directly from the definition of O-notation. (Do not use the theorem on polynomial orders.)

9x3-11x2+3x is O(x3)

5. (4 pts.) Show that the function g:R® R defined by the rule g(x)=-(x/4)-8 is decreasing on the set of all real numbers.