If `a, b, c` are in `A.P and a, b, d` are in `G.P,` prove that `a, a -b, d -c` are in `G.P.`

Answer

N/a

Explanation

a, b, c are in A.P.
`rArr" "b=(a+c)/(2)`
a, b, d are in G.P.
`rArr" "b^(2)=ad`
To prove : a, a-b, d-c are in G.P.
`rArr" "(a-b)^(2)=a(d-c)`
`R.H.S.=a(d-c)`
`=a[(b^(2))/(a)-(2b-a)]" [From eqs. (1) and (2)]"`
`=a((b^(2))/(a)-2b+a)=b^(2)-2ab+a^(2)=(a-b^(2))`
`L.H.S.`