Background information:

Notation: In this homework, we use the following notations: . In denotes the left Riemann sum using n rectangles . R. denotes the right Riemann sum using n rectangles . M. denotes the midpoint Riemann sum using n rectangles2. (6 points) Write the following sums in expanded form: 5 (a) (2 points) >vi i= 1 n-1 (b) (2 points) _(-1) j=0 (c) (2 points) ) 2x* * = 5 3. (5 points) Prove that rin - T - : Hint: What can you say about (r - 1) ) ri-l? i=1 4. (10 points) Remember that Si=(n+in n(n + 1)(2n + 1) n 13 = n'(n + 1)2 i=1 2 i= 1 6 4 i= 1 Use these formulas to compute the value of the following sums. Note that you may also need to use other properties of sums. 12 (a) (2 points) _(k - 2) * = 1 10 (b) (4 points) > (3i + 3) i= 1 20 (c) (4 points) _(2m2 - 5) m=3