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Question Number 151426 by peter frank last updated on 21/Aug/21

Express (1/(5×9)) in partial fraction

Express15×9inpartialfraction

Answered by liberty last updated on 21/Aug/21

(1/(n(n+4)))=(a/n)+(b/(n+4)) a= [(1/(n+4)) ]_(n=0) =(1/4) b= [(1/n) ]_(n=−4) =−(1/4) (1/(n(n+4)))=(1/(4n))−(1/(4(n+4))) n=5⇒(1/(5×9)) =(1/(20))−(1/(36))

1n(n+4)=an+bn+4a=[1n+4]n=0=14b=[1n]n=−4=−141n(n+4)=14n−14(n+4)n=5⇒15×9=120−136

Answered by MJS_new last updated on 21/Aug/21

it′s not unique (1/((a−b)(a+b)))=(1/(2(a−b)b))−(1/(2(a+b)b))= [a=7∧b=2] =(1/(20))−(1/(36)) but (1/((a−b)(a+b)))=(1/(2a(a+b)))+(1/(2a(a−b)))= [a=7∧b=2] =(1/(126))+(1/(70))

it′snotunique1(a−b)(a+b)=12(a−b)b−12(a+b)b=[a=7∧b=2]=120−136but1(a−b)(a+b)=12a(a+b)+12a(a−b)=[a=7∧b=2]=1126+170

Answered by MJS_new last updated on 21/Aug/21

(1/(5×9))=(1/a)+(1/b) with a≥b ((a),(b) ) ∈{ (((18)),((−30)) ) , (((20)),((−36)) ) , (((30)),((−90)) ) , (((36)),((−180)) ) , (((40)),((−360)) ) , (((42)),((−630)) ) , (((44)),((−1980)) ) , (((90)),((90)) ) , (((120)),((72)) ) , (((126)),((70)) ) , (((180)),((60)) ) , (((270)),((54)) ) , (((450)),((50)) ) , (((720)),((48)) ) , (((2070)),((46)) ) }

15×9=1a+1bwitha⩾b(ab)∈{(18−30),(20−36),(30−90),(36−180),(40−360),(42−630),(44−1980),(9090),(12072),(12670),(18060),(27054),(45050),(72048),(207046)}

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