Find four numbers in G.P. whose sum is 85 and product is 4096.

Explanation

Let the four numbers in G.P. be `a/r^(3),a/r,ar,ar^(3)`
Product=`a^(4)=4096=8^(4)`
`rArra=8`
Sum=`8(1/(r^(3))+1/r+r+r^(3))=85`
`rArr8(r^(3)+1/(r^(3)))+8(r+1/r)-85=0`
`rArr8[(r+1/r)^(3)-3(r+1/r)]+8(r+1/r)-85=0` …(1)
Let `r+1/r=t`
So, (1) reduces to
`8t^(3)-16t-85=0`
Put 2t=y
`thereforey^(3)-8y-85=0` ....(2)
Put 2t=y
`therforey^(3)-8y-85=0`
`rArr(y-5)(y^(2)+5y+17)=0`
`thereforey=5` (The other factor gives imaginary values)
`rArr2(r+1/r)=5`
`rArr2r^(2)-5r+2=0`
`rArr(r-2)(2r-1)=0`
`rArrr=2,1/2anda=8`
Hence, the four numbers are 1,4,16,64 or 64,16,4,1.