questions 3-7

V3 V5 V7 V21 3. Consider the sum (3)2 + (8)2 + (13)2 + . .. + (15n - 2)2: (a) Write the sum in sigma notation. (b) Use identities _ k = , [m(m + 1)] and >k? = = [m(m + 1)(2m + 1)] to prove that k = 1 K = 1 (3) 2 + (8) 2 + (13) 2 + ... + (15n - 2)2 - 5(n) (450n2 + 69n - 35). 4. Prove that e(e + 1) = 2 [n(n + 1)(2n + 1)] by each of the following two methods: (a) By mathematical induction on positive integer n 2 1. ( b ) By using the identities mentioned in part (b) of question 3 . 5. Evaluate 1 290 2 + (-2)91 4 -2+ V32 , where z - -V3 + 2. Simplify as much as possible. 6. For each of the following statements, if it is true prove it, and if it is false give a counter example. Z (a) 12 /2 - Z ' (2 # 0).; (b) arg ( z + z) = 0; e402 i (e02) 4 (c) = cos(20 + 1)2 + isin(20 + 1)2 . 7. Find all fifth roots of z = -16v2 -16v2i. Write the roots in exponential form and use principal value of their arguments