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If a, b, c ,d be in G.P. , show that ...

If a, b, c ,d be in G.P. , show that
(i) `(b -c)^(2) + (c - a)^(2) +(d -b)^(2) = (a - d)^(2)`
(ii) `a^(2) + b^(2) + c^(2) , ab + bc + cd , b^(2) + c^(2) + d^(2) ` are in G.P.

Text Solution

Verified by Experts

(i) Let r be the common ratio of the G.P.
than b = ar , ` ar^(2) and d = ar^(3)`
L.H.S = `(b-c)^(2) + (c-a)^(2) + (d-b)^(2)`
` = (ar-ar^(2))^(2) + (ar^(2) - a)^(2) + (ar^(3) - ar)^(2)`
=` a^(2) r^(2) (1-r)^(2) + a^(2) (r^(2) -1)^(2) + a^(2) r^(2) (r^(2) -1)^(2)`
`a^(2) [r^(2) (1 + r^(2) -2r) + (r^(4) + 1 - 2r^(2)) + r^(2) (r^(4) + 1 - 2r^(2))]`
=`a^(2) [r^(2) 1 + r^(4) -2r^(3) + r^(4) + 1 - 2r^(2) + r^(6) + r^(2) - 2r^(4)]`
`=a^(2)[r^(6) + 2r^(2) - 1]`
` a^(2) (r^(3) -1)^(2)`
`= [a(r^(3) -1)]^(2)`
= `(ar^(3) -a)^(2)`
` (d -a)^(2)`
` = (a -d)^(2)`
Let r be the common ratio of the G.P.
than `b = ar , c ar^(2) and d = ar^(3)`
Now, ` a^(2) + b^(2) + c^(2) = a^(2) +a^(2) r^(2) + a^(2) r^(4) = a^(2) (1 + r^(2) + r^(4))`
`ab + bc + cd = a^(2) r + a^(2) r^(3) + a^(2) t^(5) = a^(2)r(1 + t^(2) + r^(4))`
` b^(2) + c^(2)+d^(2) = a^(2)r^(2) + a^(2) r^(4) + a^(2) t^(6) = a^(2)r^(2)(1 + t^(2) + r^(4))`
Clearly , `(ab + bc + cd)/(a^(2)+ b^(2) + c^(2))=(b^(2) + c^(2)+d^(2))/(ab + bc + cd)`
Hence , ` a^(2) + b^(2) + c^(2) , ab + bc + cd , b^(2) + c^(2) + d^(2)` are in G.P.
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