1

### IIT-JEE 2007

MCQ (Single Correct Answer)
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
$${Q_1} = \,\,{Q_2} = \,\,{Q_3} = ...$$
2

### IIT-JEE 2007

MCQ (Single Correct Answer)
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 $${A_1} > {A_2}\, > {A_3} > ...$$
3

### IIT-JEE 2007

MCQ (Single Correct Answer)
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
$${H_1} > {H_2}\, > {H_3} > ...$$
B
$${H_1} < {H_2}\, < {H_3} < ...$$
C
$${H_1} > {H_2}\, > {H_3} > ...$$ and $${H_1} < {H_2}\, < {H_3} < ...$$
D
$${H_1} < {H_2}\, < {H_3} < ...$$ and $${H_1} > {H_2}\, > {H_3} > ...$$
4

### IIT-JEE 2007

MCQ (Single Correct Answer)
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 $${T_r} = \,{V_{r + 1}} - \,{V_r} - 2\,\,\,and\,\,\,{Q_r} = \,{T_{r + 1}} - \,{T_r}\,for\,r = 1,2,...$$

$${T_r}$$ is always

A
an odd number
B
an even number
C
a prime number
D
a composite number

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