1.

If `a ,b ,c ,d`are in G.P., prove that `a+b+,b+c ,c+d`are also in G.P.

Answer» Let a, b, c, d be in GP with common ratio r. Then,
`b=ar, c=ar^(2)` and `d=ar^(3)`.
`:. (a+b)=(a+ar)=a(1+r),
(b+c)=(ar+ar^(2))=ar(1+r)`,
`(c+d)=(ar^(2)+ar^(3))=ar^(2) (1+r)`.
`:. (b+c)^(2)=a^(2)r^(2)(1+r)^(2)` and `(a+b)(c+d)=a^(2)r^(2)(1+r)^(2)`.
Consequently, `(b+c)^(2)=(a+b)(c+d)`.
Hence, `(a+b), (b+c)` and `(c+d)` are in GP.


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