Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 12

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If $1^{5} + 2^{5} + 3^{5} + . . . . . . . . . . . . + n^{5}$ is divided by $1^{3} + 2^{3} + 3^{3} + . . . . . . . . . . . . . + n^{3}$ , the remainder is $(when n = 3 m + 1)$

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Important Questions on Sequences and Series

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Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 13

The numbers

x_{1}, x_{2}, x_{3} . . . . . .

form an infinite decreasing

G . P .

If

x_{1} = 1

, then the common ratio of the progression for which the expression

6 x_{5} - 16 x_{4} - 3 x_{3} + 12 x_{2}

is maximum, is

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Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 14

If

w, x, y, z \in R^{+}

and

256 x y w z \geq {(w + x + y + z)}^{4}

and

3 w + 5 x + 7 y + 9 z = 24

, then the roots of the equation

(w + x) t^{2} + y z t + \sqrt{x + y + z + w} = 0

are

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Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 15

Solve the following equation for $x$ and $y$ :

$\log_{10} x + \frac{1}{2} \log_{10} x + \frac{1}{4} \log_{10} x + . . . . . . . . = y$ , $\frac{1 + 3 + 5 + . . . . . . (2 y - 1)}{4 + 7 + 10 . . . . . . . (3 y + 1)} = \frac{20}{7 \log_{10} x},$ then $y$ is

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Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 16

If

n

is a root of

x^{2} (1 - a c) - x (a^{2} + c^{2}) - (1 + a c) = 0

and if

n

harmonic means are inserted between

a

and

c

, then the difference between first and last mean is

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Gamma Question Bank for Engineering Mathematics>Chapter 4 - Sequences and Series>Assignment>Q 9

Let

2048

arithmetic means be inserted between

2^{22} - 1

and

2^{22} + 1

. Suppose the sum of these arithmetic means is

S

. Then

\frac{S}{2^{33}}

is ....