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Question-29-Theres-2-same-ratio-sequences-a-n-b-n-with-a-ratio-of-non-zero-And-if-two-sums-of-each-sequences-n-1-a-n-n-1-b-n-are-convergent-and-two-equations-n-1-a-n-b




Question Number 213298 by MathematicalUser2357 last updated on 02/Nov/24
Question 29. Theres 2 same-ratio sequences {a_n },{b_n } with a ratio of non-zero  And if two sums of each sequences (Σ_(n=1) ^∞ a_n ,Σ_(n=1) ^∞ b_n ) are convergent, and two equations  Σ_(n=1) ^∞ a_n b_n =(Σ_(n=1) ^∞ a_n )×(Σ_(n=1) ^∞ b_n ) and 3×Σ_(n=1) ^∞ ∣a_(2n) ∣=7×Σ_(n=1) ^∞ ∣a_(3n) ∣ are true.  If Σ_(n=1) ^∞ ((b_(2n−1) +b_(3n+1) )/b_n )=S, then Find the value of 120S.  (korea university exam question)
Question29.Theres2sameratiosequences{an},{bn}witharatioofnonzeroAndiftwosumsofeachsequences(n=1an,n=1bn)areconvergent,andtwoequationsn=1anbn=(n=1an)×(n=1bn)and3×n=1a2n∣=7×n=1a3naretrue.Ifn=1b2n1+b3n+1bn=S,thenFindthevalueof120S.(koreauniversityexamquestion)
Answered by MrGaster last updated on 02/Nov/24
Σ_(n=1) ^∞ a_n =(a_1 /(1−r_a )),Σ_(n=1) ^∞ b_n =(b_1 /(1−r_b ))  Σ_(n=1) ^∞ a_n b_n =((a_1 b_1 )/(1−r_a r_b ))  ((a_1 b_1 )/(1−r_a r_b ))=(a_1 /(1−r_a ))∙(b_1 /(1−r_b ))  3Σ_(n=1) ^∞ ∣a_(2n) ∣=7Σ_(n=1) ^∞ ∣a_(3n) ∣  3((∣a_1 ∣r_a )/(1−r_a ^2 ))=7((∣a_1 ∣r_a ^2 )/(1−r_a ^3 ))  3(1−r_a ^3 )=7(1−r_a ^2 )  3−3r_a ^3 −7r_a ^2 +4=0  (r_a −1)(3r_a ^2 −4r_a −4)=0  ra=1,r_a =2,r_a =−(2/3)  r_a =1 does not satisfy the condlition for  convergence.since∣r_a ∣<1, have  r_a =−(2/3).  Σ_(n=1) ^∞ ((b_(2n−1) +b_(3n+1) )/b_n )=Σ_(n=1) ^∞ (((b_1 r_b ^(2n−2) +b_1 r_b ^(3n) )/(b_1 r_b ^(n−1) )))  =Σ_(n=1) ^∞ (((r_b ^(2n−2) +b_1 r_b ^(3n) )/r_b ^(n−1) ))  =Σ_(n=1) ^∞ (r_b ^(n−1) +r_b ^(2n+1) )  =Σ_(n=1) ^∞ r_b ^(n−1) +Σ_(n=1) ^∞ r_b ^(2n+1)   =(1/(1−r_b ))+(r_b ^3 /((1−r_b )(1+r_b )))  =((1+r_b +r_b ^3 )/(1−r_b ^2 ))  =((1+r_b +r_b ^3 )/((1−r_b )(1+r_b )))  =((1+r_b +r_b ^3 )/(1−r_b ^2 ))  Given that 120S=120(((1+r_b +r_b ^3 )/(1−r_b ^2 ))).lf r_b =−(2/3)  120S=120(((1−(2/3)+(−(2/3))^3 )/(1−(−(2/3))^2 )))  =120(((1−(2/3)−(8/(27)))/(1−(4/9))))  =120(((27−18−8)/((27)/(5/9))))  =120((1/((27)/(5/8))))  =120((1/(27))∙(9/5))  =120∙(1/(15))              determinant (((=8)))
n=1an=a11ra,n=1bn=b11rbn=1anbn=a1b11rarba1b11rarb=a11rab11rb3n=1a2n∣=7n=1a3n3a1ra1ra2=7a1ra21ra33(1ra3)=7(1ra2)33ra37ra2+4=0(ra1)(3ra24ra4)=0ra=1,ra=2,ra=23ra=1doesnotsatisfythecondlitionforconvergence.sincera∣<1,havera=23.n=1b2n1+b3n+1bn=n=1(b1rb2n2+b1rb3nb1rbn1)=n=1(rb2n2+b1rb3nrbn1)=n=1(rbn1+rb2n+1)=n=1rbn1+n=1rb2n+1=11rb+rb3(1rb)(1+rb)=1+rb+rb31rb2=1+rb+rb3(1rb)(1+rb)=1+rb+rb31rb2Giventhat120S=120(1+rb+rb31rb2).lfrb=23120S=120(123+(23)31(23)2)=120(123827149)=120(271882759)=120(12758)=120(12795)=120115=8

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