JEE Advance - Mathematics (2008 - Paper 2 Offline - No. 16)
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. and
STATEMENT-2 The numbers $${b_1},\,{b_{2\,}},\,{b_3},\,{b_4}\,$$ are in H.P.
STATEMENT-1 is True, STATEMENT-2 is True;
STATEMENT-2 is a correct explanation for
STATEMENT-1
STATEMENT-1 is True, STATEMENT-2 is True;
STATEMENT-2 is NOT a correct explanation for
STATEMENT-1
STATEMENT-1 is True, STATEMENT-2 is False
STATEMENT-1 is False, STATEMENT-2 is True
Explanation
Given, $$a_1,a_2,a_3,a_4$$ are in G.P.
Then, $$b_1,b_2,b_3,b_4$$ are the numbers.
$$a_1,a_1+a_2,a_1+a_2+a_3,a_1+a_2+a_3+a_4$$ or $$a,a+ar,a+ar+ar^2,a+ar+ar^2+ar^3$$
Clearly above numbers are neither in A.P. nor in G.P. and hence statement 1 is true.
Also, $${1 \over a},{1 \over {a + ar}},{1 \over {a + ar + a{r^2}}},{1 \over {a + ar + a{r^2} + a{r^3}}}$$ are not in H.P.
$$\therefore$$ $$b_1,b_2,b_3,b_4$$ are not in H.P.
$$\therefore$$ Statement 2 is false.
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