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Sequences and Series

Multiple Choice Questions

111.

If the arithmetic mean of a and b is , then the value of n is

- 1
0
1
None of the above

112.

equals

e + 1
$\frac{e - 1}{e + 1}$
e - 1
None of these

113.

The sum of the series

$(\frac{a - b}{a}) + \frac{1}{2} {(\frac{a - b}{a})}^{2} + \frac{1}{3} {(\frac{a - b}{a})}^{3} + . . . \infty$ is

$\log_{e} (\frac{a}{b})$
$\log_{e} (\frac{a - b}{a})$
$\log_{e} (\frac{b}{a})$
None of these

114.

The harmonic mean of the roots of the equation

$(5 + \sqrt{2}) x^{2} - (4 + \sqrt{5}) x + 8 + 2 \sqrt{5} = 0$ is

115.

The sum of n terms of the following series 1 + (1 + x) + (1 + x + x²) +... will be

$\frac{1 - x^{n}}{1 - x}$
$\frac{x (1 - x^{n})}{1 - x}$
$\frac{n (1 - x) - x (1 - x^{n})}{{(1 - x)}^{2}}$
None of the above

116.

If A₁, A_2; G₁, G₂ and H₁, H₂ be two AM's, GM's and HM's between two quantities, then the value of $\frac{G_{1} G_{2}}{H_{1} H_{2}}$ is

$\frac{A_{1} + A_{2}}{H_{1} + H_{2}}$
$\frac{A_{1} - A_{2}}{H_{1} + H_{2}}$
$\frac{A_{1} + A_{2}}{H_{1} - H_{2}}$
$\frac{A_{1} - A_{2}}{H_{1} - H_{2}}$

117.

1² + 1 + 2² + 2 + 3²+ 3 + ... + n² + n is equal to

$\frac{n (n + 1)}{2}$
${[\frac{n (n + 1)}{2}]}^{2}$
$\frac{n (n + 1) (n + 2)}{3}$
$\frac{n (n + 1) (n + 2) (n + 3)}{4}$

118.

2^1/4 . 4^1/8 . 8^1/16 . 16^1/32 ... is equal to

1
2
$\frac{3}{2}$
$\frac{5}{2}$

119.
The sum of the series
$\sum {(- 1)}^{r} {}^{n}C_{r} (\frac{1}{2^{r}} + \frac{3^{r}}{2^{2 r}} + \frac{7^{r}}{2^{3 r}} + \frac{15^{r}}{2^{4 r}} + . . . m terms)$ is

$\frac{2^{mn} - 1}{2^{mn} (2^{n} - 1)}$

$\frac{2^{mn} - 1}{2^{n} - 1}$

$\frac{2^{mn} + 1}{2^{n} + 1}$

None of these

$\frac{2^{mn} - 1}{2^{mn} (2^{n} - 1)}$

$\sum {(- 1)}^{r} {}^{n}C_{r} (\frac{1}{2^{r}} + \frac{3^{r}}{2^{2 r}} + \frac{7^{r}}{2^{3 r}} + \frac{15^{r}}{2^{4 r}} + . . . m terms) = \sum_{r = 0}^{n} {(- 1)}^{r} . {}^{n}C_{r} \frac{1}{2^{r}} + \sum_{r = 0}^{n} (- 1) . {}^{n}C_{r} \frac{3^{r}}{2^{2 r}} + \sum_{r = 0}^{n} (- 1) . {}^{n}C_{r} \frac{7^{r}}{2^{3 r}} + . . . m terms = {(1 - \frac{1}{2})}^{n} + {(1 - \frac{3}{4})}^{n} + {(1 - \frac{7}{8})}^{n} + . . . m terms = \frac{1}{2^{n}} + \frac{1}{4^{n}} + \frac{1}{8^{n}} + . . . m terms = \frac{\frac{1}{2^{n}} (1 - \frac{1}{2^{mn}})}{(1 - \frac{1}{2^{n}})} = \frac{2^{mn} - 1}{2^{mn} (2^{n} - 1)}$