Suppose four distinct positive
numbers a

_{1}, a_{2}, a_{3},a_{4 }are in G.P. Let b_{1}= a_{1}, b_{2}= b_{1}+a_{2}, b_{3}= b_{2}+a_{3}and b_{4}= b_{3}+a_{4.}
Statement-1: The numbers b

_{1}, b_{2}, b_{3},b_{4 }are neither in A.P. nor in G.P._{ }
Statement-2: The numbers b

_{1}, b_{2}, b_{3},b_{4 }are in H.P.
(a) Statement-1 is True, Statement-2
is True; Statement-2 is a correct explanation for statement-1;

(b) Statement-1 is True, Statement-2 is True;
Statement-2 is not a correct explanation for statement-1;

(c) Statement-1 is True, Statement-2
is False;

(d) Statement-1 is False, Statement-2
is True;

_{}
Answer follows here:

__Solution:__
As
a

_{1}, a_{2}, a_{3},a_{4 }are in G.P., let a_{1}= a, a_{2}= ar, a_{3}= ar^{2}, a_{4}= ar^{3}
b

_{1}= a_{1}= a, b_{2}= b_{1}+a_{2}= a+ar, b_{3}= b_{2}+a_{3}= a+ar+ar^{2}, b_{4}= b_{3}+a_{4}= a+ar+ar^{2}+ar^{3}__Let us check for A.P.:__

b

_{2}-b_{1}= a+ar-a = ar, b_{3}-b_{2}= a+ar+ar^{2}-a-ar = ar^{2}=> b_{2}-b_{1}≠ b_{3}-b_{2 }=> b_{1}, b_{2}, b_{3},b_{4 }are not in A.P.__Let us check for G.P.:__

b

_{2}/b_{1}= (a+ar)/a = 1+r, b_{3}/b_{2}= (a+ar+ar^{2})/(a+ar) = (1+r+r^{2})/(1+r) => b_{2}/b_{1}≠ b_{3}/b_{2 }=> b_{1}, b_{2}, b_{3},b_{4 }are not in G.P.__Let us check for H.P.:__

(1/b

_{2})-(1/b_{1}) = (1/a+ar)-(1/a) = -r/(a+ar), (1/b_{3})-(1/b_{2}) = (1/a+ar+ar^{2})-(1/a+ar) = -r^{2}/a(1+r)(1+r+r^{2}) => 1/b_{1}, 1/b_{2}, 1/b_{3},1/b_{4 }are not in A.P._{ }=> b_{1}, b_{2}, b_{3},b_{4 }are not in H.P.**Answer (c)**
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