Sequence And Series Question 577
Question: If the $ 4^{th},\ 7^{th} $ and $ 10^{th} $ terms of a G.P. be $ a,\ b,\ c $ respectively, then the relation between $ a,\ b,\ c $ is
[MNR 1995; Karnataka CET 1999]
Options:
A) $ b=\frac{a+c}{2} $
B) $ a^{2}=bc $
C) $ b^{2}=ac $
D) $ c^{2}=ab $
Show Answer
Answer:
Correct Answer: C
Solution:
Let first term of G.P. $ =A $ and common ratio $ =r $ We know that $ n^{th} $ term of G.P. = $ A{r^{n-1}} $ Now $ t_4=a=Ar^{3},\ t_7=b=Ar^{6} $ and $ t_{10}=c=Ar^{9} $ Relation $ b^{2}=ac $ is true because $ b^{2}={{(Ar^{6})}^{2}}=A^{2}r^{12} $ and $ ac=(Ar^{3})(Ar^{9})=A^{2}r^{12} $ Aliter: As we know, if $ xy+2y^{2}+yz=xy+xz+y^{2}+yz $ in A.P., then $ =2n^{2}+5n-2n^{2}+4n-2-5n+5=4n+3 $ terms of a G.P. are always in G.P., therefore, $ a,\ b,\ c $ will be in G.P. $ i.e. $ $ 2,\ 5,\ 8,\ 11,\ 14=40 $ .
BETA
