Lesson.6 Exercises, pges 68 7 A. Determine whether ech infinite geometric series hs finite sum. How do you know? ) + +.5 + 6.75 +... r is:.5, so the sum is not finite. b) 0.5 0.05 0.005 0.0005... r is: 0.05 0.5 0., so the sum is finite. c) 8 + 7 8 +Á r is:, so the sum is finite. 8 d) 0. + 0. + 0. + 0.8 +... 0. r is:, so the sum is not finite. 0.. Write the first terms of ech infinite geometric series. ) =, r = 0. b) =, r =0.5 t is: (0.). t is: (0.5) 0.5 t is:.(0.) 0.6 t is: 0.5(0.5) 0.065 t is: 0.6(0.) 0.08 t is: 0.065(0.5) 0.05 65 c) =, r = d) =, r = 5 8 t is: 5 b 5 t is: 5 5 b 5 t is: 5 5 b 5 t is: 8 b 6 t is: 7 6 8 b 8 7 t is: 8 8 8 b 0 P DO NOT COPY..6 Infinite Geometric Series Solutions
5. Ech infinite geometric series converges. Determine ech sum. 7 ) 8 + + 0.5 + 0.5 +... b) 6 6... Use: Use: S q Substitute: 8, r Substitute:, r 8 0.6 The sum is 0.6. The sum is. 0 0 80 c) 0 + 7 +... d) + + 7... Use: Use: S q Substitute: 0, r Substitute:, r 0 b b 6 The sum is 6..5 The sum is.5. B 6. Wht do you know bout the common rtio of n infinite geometric series whose sum is finite? The common rtio is less thn nd greter thn. 7. Use the given dt bout ech infinite geometric series to determine the indicted vlue. ) =, S q = 6; determine r b) r =, S q = 7 ; determine Substitute for nd S q Substitute for r nd in. in S q. t 6 7 6 6r b 6r r, or 7 t 7 b 6 6 S q.6 Infinite Geometric Series Solutions DO NOT COPY. P
8. Use n infinite geometric series to express ech repeting deciml s frction. ) 0.7 b). 0.7 0. 0.07. 0. 0.000 7... 0.000... This is n infinite This is n infinite geometric series with geometric series with 7 nd nd 000 000 0.000 7 0.000 r is r is 0.07 00 0. 000 Substitute for nd r Substitute for nd r in. in. 7 000 000 00 000 7 000 7 000, or, or 0 00 000 7 Add: Add: 0 0 0 So, 0.7 So,. 0. The hour hnd on clock is pointing to. The hnd is rotted clockwise 80, then nother 60, then nother 0, nd so on. This pttern continues. ) Which number would the hour hnd pproch if this rottion continued indefinitely? Explin wht you did. The ngles, in degrees, tht the hnd rottes through form geometric sequence with 80 nd r. The totl ngle turned through is the relted infinite geometric series: 80 80 80 80 + + + +... Use: Substitute: 80, r 80 70 When the hour hnd hs rotted 70º clockwise from, it will point to. So, if the rottion continued indefinitely, the hour hnd would pproch. b) Wht ssumptions did you mke? I ssumed tht the ngle mesures formed n infinite geometric series tht converged. P DO NOT COPY..6 Infinite Geometric Series Solutions 5
0. Brd hs blnce of $500 in bnk ccount. Ech month he spends 0% of the blnce remining in the ccount. ) Express the totl mount Brd spends in the first months s series. Is the series geometric? Explin. After: Amount spent Amount remining month $500(0.) $00 $500(0.6) $00 months $00(0.) $0, $00(0.6) $80, or $500(0.6)(0.) or $500(0.6) months $80(0.) $7, $80(0.6) $08, or $500(0.6) (0.) or $500(0.6) months $08(0.) $.0, or $500(0.6) (0.) The mounts spent re: $500(0.) $500(0.6)(0.) $500(0.6) (0.) $500(0.6) (0.) This is geometric series with $500(0.) nd r 0.6 b) Determine the pproximte mount Brd spends in 0 months. Explin wht you did. The mount Brd spends in 0 months is the sum of the firs0 terms of the series in prt. ( n ) Use: S n, r Substitute: n 0, 00, r 0.6 00( 0.6 0 ) S 0 0.6 S 0 6.766... Brd spends bout $6.8 in 0 months. c) Suppose Brd could continue this pttern of spending indefinitely. Would he eventully empty his bnk ccount? Explin. No, becuse Brd cn only spend money in dollrs nd cents, nd not frctions of cent, so ech mount he spends will be rounded to the nerest cent. Continuing the pttern of spending 0% ech month, nd rounding to the nerest cent ech time, Brd will eventully end up with $0.0 remining in his ccount. Since 0% of $0.0 is less thn penny, this mount will never be spent. 6.6 Infinite Geometric Series Solutions DO NOT COPY. P
C. Write the product of 0. nd 0. b s frction, where nd b represendigit nturl numbers. Explin your strtegy. 0. 0. + 0.0 + 0.00 +... This is n infinite geometric series with 0., or nd 0 r 0., or. 0 Use: Substitute:, r 0 0 0 0 b 0 0 b So, 0. ; similrly, 0. b ; then (0. )( 0. b) is b bb b 8. Crete different infinite geometric series with sum of. Explin wht you did. Smple response: Choose vlue for r between nd, then determine vlue for. Let r 0.5. Let r 0.6. S q Use: Use: Substitute: Substitute:, r 0.5, r 0.6 (0.5) 0.6 5.6 5 5 5 One series is: 5... 6 6 Another series is:.6 0.6 0.576 0.56... S q P DO NOT COPY..6 Infinite Geometric Series Solutions 7
. Determine the sum of this infinite geometric series: + + + + + Á 8 8 5 8 The common rtio is r:, or Use: Substitute:, r The sum is. 8.6 Infinite Geometric Series Solutions DO NOT COPY. P