Chapter 19

  1. 1.

    1. a1 = 8 , d =  − 1/2 ;  arith. seq. 8, 15/2, 7, 13/2, 6, …

    2. a1 = 8 , r =  − 1/2 ;  geom. seq. 8,  − 4 , 2 ,  − 1 , 1/2 ,  … 

  2. 2. 6,  − 2 ,  23 ,  …  ; 

    geometric sequence

    a1 = 6r =  − 26 =  − 13S7 = 6 [ 1 − ( − 13)7 ] 1 − ( − 13) = 6(1 + 137)43 = 1094243

  3. 3. d = (x + 6) − 4andd = (3x + 2) − (x + 6)d = x + 2andd = 2x − 4

    So x + 2 = 2x − 4x = 6

    Thus ,  d = 8.a10 = 4 + 9(8) = 76S10 = 102(4 + 76) = 400

  4. 4. 0.454545 …  = 0.45 + 0.0045 + 0.000045 +  ⋯ a = 0.45r = 0.01S = 0.451 − 0.01 = 0.450.99 = 511

  5. 5. 1 − 4x = (1 − 4x)1/2 = 1 + (12)( − 4x) + 12(12 − 1)2( − 4x)2 +  ⋯  = 1 − 2x − 2x2 +  ⋯ 

  6. 6. (2x − y)5 = (2x)5 + 5(2x)4( − y) + 5(4)2(2x)3( − y)2 + 5(4)(3)2(3)(2x)2( − y)3 + 5(4)(3)(2)2(3)(4)(2x)( − y)4 + ( − y)5 = 32x5 − 80x4y + 80x3y2

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