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1

### IIT-JEE 2008

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.

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
2

### IIT-JEE 2007

Let $${A_1}$$, $${G_1}$$, $${H_1}$$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For $$n \ge 2,\,Let\,{A_{n - 1}}\,\,and\,\,{H_{n - 1}}$$ have arithmetic, geometric and harminic means as $${A_n},{G_n}\,,{H_n}$$ repectively.

Which one of the following statements is correct ?

A
$${G_1} > {G_2}\, > {G_3} > ...$$
B
$${G_1} < {G_2}\, < {G_3} < ...$$
C
$${G_1} = {G_2}\, = {G_3} = ...$$
D
$${G_1} < {G_2}\, < {G_3} < ...$$ and $${G_1} > {G_2}\, > {G_3} > ...$$
3

### IIT-JEE 2007

Let $$\,{V_r}$$ denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r-1). Let $${T_r} = \,{V_{r + 1}} - \,{V_r} - 2\,\,\,and\,\,\,{Q_r} = \,{T_{r + 1}} - \,{T_r}\,for\,r = 1,2,...$$

Which one of the following is a correct statement?

A
$${Q_1},\,\,{Q_2},\,\,{Q_3},...$$ are A.P. with common difference 5
B
$${Q_1},\,\,{Q_2},\,\,{Q_3},...$$ are A.P. with common difference 6
C
$${Q_1},\,\,{Q_2},\,\,{Q_3},...$$ are A.P. with common difference 11
D
$${Q_1} = \,\,{Q_2} = \,\,{Q_3} = ...$$
4

### IIT-JEE 2007

Let $${A_1}$$, $${G_1}$$, $${H_1}$$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For $$n \ge 2,\,Let\,{A_{n - 1}}\,\,and\,\,{H_{n - 1}}$$ have arithmetic, geometric and harminic means as $${A_n},{G_n}\,,{H_n}$$ repectively.

Which one of the following statements is correct ?

A
$${A_1} > {A_2}\, > {A_3} > ...$$
B
$${A_1} < {A_2}\, < {A_3} < ...$$
C
$${A_1} > {A_2}\, > {A_3} > ...$$ and $${A_1} < {A_2}\, < {A_3} < ...$$
D
$${A_1} < {A_2}\, < {A_3} < ...$$ and $${A_1} > {A_2}\, > {A_3} > ...$$

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