The sequence `a_(1),a_(2),a_(3),".......,"` is a geometric sequence with common ratio r. The sequence `b_(1),b_(2),b_(3),".......,"` is also a geometr

Question

The sequence `a_(1),a_(2),a_(3),".......,"` is a geometric sequence with common ratio r. The sequence `b_(1),b_(2),b_(3),".......,"` is also a geometr

The sequence `a_(1),a_(2),a_(3),".......,"` is a geometric sequence with common ratio r. The sequence `b_(1),b_(2),b_(3),".......,"` is also a geometric sequence. If `b_(1)=1,b_(2)=root4(7)-root4(28)+1,a_(1)=root4(28)" and "sum_(n=1)^(oo)(1)/(a_(n))=sum_(n=1)^(oo)(1)/(b_(n))`, then the value of `(1+r^(2)+r^(4))` is

Monjur Alahi

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2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

`a_(1),a_(2),a_(3),".......,"`are in GP with common ratio r
and `b_(1),b_(2),b_(3),".......,"` is also a GP i.e. `b_(1)=1`
`b_(2)=root4(7)-root4(28)+1,a_(1)=root4(28)"
and "sum_(n=1)^(oo)(1)/(a_(n))=sum_(n=1)^(oo)(1)/(b_(n))`
`(1)/(a_(1)),(1)/(a_(2)),(1)/(a_(3)),"+.........+"oo=b_(1)+b_(2)+b_(3)+"......."+oo`
`=(1)/(root4(28))+(1)/(root4(28)r)+(1)/(root4(28)r^(2))+"......."+oo`
`=1+(root4(7)-root4(28)+1)+(root4(7)-root4(28)+1)^(2)+"......."+oo`
`implies((1)/(root4(28)))/(1-(1)/(r ))=(1)/(1-root4(7)+root4(28)-1)`
`implies(r )/((r-1)root4(28))=(1)/(root4(7)+(root4(4)-1))`
`implies(r )/(r-1)(1)/root4(4)=(1)/(root4(4-1))`
Let `root4(4)=alpha`, we get
`implies(r )/((r-1)alpha)=(1)/(alpha-1)`
`implies ralpha-r =ralpha-alpha implies r=alpha`
`implies r=root4(4)`
Now, `1+r^(2)+r^(4)=1+(root4(4))^(2)+(root4(4))^(4)`
`=1+4^((1)/(2))+4=1+2+4=7`.