IIT-JEE 2007 | Sequences and Series Question 25 | Mathematics

1

IIT-JEE 2007

MCQ (Single Correct Answer)

+4

-1

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 $${G_1} > {G_2}\, > {G_3} > ...$$

2

IIT-JEE 2007

MCQ (Single Correct Answer)

+4

-1

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

$${Q_1} = \,\,{Q_2} = \,\,{Q_3} = ...$$

3

IIT-JEE 2007

MCQ (Single Correct Answer)

+4

-1

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)

+4

-1

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 $${T_r} = \,{V_{r + 1}} - \,{V_r} - 2\,\,\,and\,\,\,{Q_r} = \,{T_{r + 1}} - \,{T_r}\,for\,r = 1,2,...$$

The sum $${V_1}$$+$${V_2}$$ +...+$${V_n}$$ is

A

$${1 \over {12}}n(n + 1)\,(3{n^2} - n + 1)$$

B

$${1 \over {12}}n(n + 1)\,(3{n^2} + n + 2)$$

C

$${1 \over 2}n(2{n^2} - n + 1)$$

D

$${1 \over 3}(2{n^3} - 2n + 3)$$