Question

Chapter 9, Section 9.2, Question 038 Use the properties of geometric series to find the sum of the given series. For what value of the variable does the series converge to this sum? 6 -- 362 +21622-129623 + ... Enclose numerators, and denominators in parentheses. For example, (a - b)/(1+ n). The sum of the given series is: QEZ The given series converges for O<zs! Oso O:> O-<< 0-05:<

Answer 1

n=1 n-1 Ans Given series = 6-362 +216 22-1296 23+ - = 6'zo - 6²2' +63 226423 aton zal 8 1"-6" ₂n = 68 (-62)n-1 n=1 = 6x Infinite geometric series § (462)n-1 = 6 A • This agcome me series and firstterm ael and common ranio cr=-62 and an eo geome mic conuerges only when Ir1 <1 and then suma a => -62</ 121<1 = -1 < zet NE 1-r =) 12/16 6 6 and sum = (2416 if JĄ Total vum = 6A = 6 1 +62 1+62 So sum= 6 Thegiven senes conuerges for -! <2C1 1+62 Ophono Pas 2 Billinvested $ 280 at start of each month. The investment yields 0.4% per month compounded annually Given To find The value after 20 months After Ist month yield = 0.4% of 280 = 280 X0.004 = 1.12 After ist month invested amount = 280 + 280x0.004 280X (1, 004) Billadds another 280 dollars so net investment = 280+280 X (1.004) = 280 (1 +1.004) --

After 2nd month yeld obtained: 0.4% of 2.80 (1+1,004): 280 (14.004) (0.004) After 2nd month invested amount becomes 280 (1+1.004) +280(1+1,004) 10.004) = 280 (1 +1.004)(1.004) = 280 (11.004)2 + (1.004)) Billodds another 280 dollars = 280 +280 (1.004)? 7(1.004)) 2280 (1 + (1.004)2 + (1.004) • After 3rd month yuild becomes = 0.47 280(1+1004 (1.004)") = 280 (1+1.004 +1.004) (0.001) After 3rd month invested amount becomes = 280 (11.004)$+(1.004) 41,004) Billaddo another $280 = 280 (17 1.004 +(1,004)? +(1.004)3) following this pattern value invested innmonth (after months) 280 (1.004)° +(1.004)'+(1.004) +---+(1.004)" ) Jovalle invested after 20 months = 280 (1.004)º +(1.004)' +1.004)17 --.+(1.004)20) This represent sum of 21 terms whose first term = 280 =r=1,004 :, sum= a pnt)-1) = 280 (1.004)22-1) - 6425.75 . Common ratio 1.004-1 = 16423.75