Question

III A Sports club offers two types of membership. Offer A A membership fee of 10,000,000 L.L for the first year only , after which a yearly amount of 500,000 L.L is to be paid after the first year. Denote by S, the amount paid during n years Offer B An annual membership fee of 1,000,000 L L for the first year which increases each year by 10% after the first year Also with this offer and starting the second year , a reduction of 50,000 LL is made to the amount paid to the club Denote by Un the amount, expressed in L.L paid by a member to the club , with offer B the n year Soul = 1,000,000 1° Determine the total sum Sn paid by a member who chose type A for the first n years 2° Let(,) be the sequence defined as Vn-Un-500,000 where n is a natural number greater or equal 1 a) Express Un+1 in terms of Un . b) Show that (V) is a geometric sequence whose common ratio and first term to be determined 3° Express Vn then Un in terms of n 4° Denote by S, the amount paid to the club by a member who chose type B for the first n years Show that S 5,000,000 (1.1)-1+500,000n , where n is non-zero . 59 What is the minimum number of years a member should subscribe so that offer A become more advantageous than offer B ?

Answer 1

Given that Ci). MembenShip fee 10, 0o, 000 L.L for Bnst year わfter first Year LL ever S 00,DOo 5,00,000 LL, everd Tear So -Amount Pail aft enr n een munt Poid aftert inst Year ס15,0 1000 ID, 00,000 (5,00,000 x !) Aften g 5,Do, DOO x2 After 3rd year 10, Do '000 + (5,00,000 3) : ,, -Dmount tt be paid after (n)

offer Membeaship fee lo, o0 ,006 LL s eam Member ship by to'. evevy Hear Start int the second year , (reduction 5 b,000 LU is macte to the toteal amount UnTotal amourt a)· Amount Pad by membeơ。n Secord dear 10,00, 000 (I5,DD,DOD 6, DO, D0O Amount Pa bT member in thTrd cteum 10,DODob 11,17-5,00, Dob. After n lo,00,coo(n)"-5,00,00 year we See UnIo,0o,000 CI.1) - 1, 00,000 C)n

le haue n1,0o,000 (,)5,00,00o -for n: , ,o,co , ooo ('·') -5,00 ,000 6,00,000 vn- : ule See , to,000 C,00,b00 ㅋ10,000 hle see both the atioule apprroximately same, So ve can say thart is in Germetoic Paogresson 1 ,o,oo, 000 r,, ,,n- Un-5,00,000 3 Va 5,00,000 : : Un : Un - Vh 16,00,000 C) On 50D,000

for , n-1, sn) = 5000,000 (11.1)-))ナ 5,00,000 . 10,00,000 00,000 2 2t, 50000 . calculated ut,na , we see -the amount Paif -this foronula Hence Proved For t be bentft cal aunt (A) aou ) 2 ID