1046 CHAPTER 11 Further Topics in Algebra The following rules, used in calculus, can be proved by mathematical induction. Summation Rules an i=1 i =1 +2 +P+ n = n1n +12 2 an i=1 i 2 =12 +22 + P+ n2 = n1n +12 12n +12 6 an i=1 i 3 =13 +23 + P+ n3 = n21n +122 4 EXAMPLE 6 Using the Summation Properties and Rules Use the summation properties and rules to evaluate each series. (a) a40 i=1 5 (b) a22 i=1 2i (c) a14 i=1 12i2 - 32 SOLUTION (a) a40 i=1 5 = 40152 Property (a) with n = 40 and c = 5 = 200 Multiply. (b) a22 i=1 2i = 2 a22 i=1 i Property (b) with c = 2 and ai = i = 2 # 22122 + 12 2 Summation rule = 506 Evaluate. (c) a14 i=1 12i2 - 32 = a14 i=1 2i2 - a14 i=1 3 Property (d) with ai = 2i 2 and bi = 3 = 2 a14 i=1 i2 - a14 i=1 3 Property (b) with c = 2 and ai = i 2 = 2 # 14114 + 1212 # 14 + 12 6 - 14132 Summation rule and Property (a) = 2030 - 42 Simplify. = 1988 Subtract. S Now Try Exercises 75, 77, and 79. an i=1 i = n1n + 12 2 an i=1 i2 = n1n + 1212n + 12 6
RkJQdWJsaXNoZXIy NjM5ODQ=