Let S K = 1 + 2 + . . . + K K and ∑ j = 1 n S 2 j = n A B n 2 + C n + D where A , B , C , D ∈ N and A Has least value then

A + C + D is not divisible by D

A + B + C + D is divisible by 5

Let α and β be the roots of x 2 - x - 1 = 0 , with α > β . For all positive integers n , define a n = α n - β n α - β , n ≥ 1 b 1 = 1 and b n = a n - 1 + a n + 1 , n ≥ 2 . Then which of the following options is/are correct?

b n = α n + β n for all n ≥ 1

MEDIUM

Earn 100

If $a, b, c, d, e$ be 5 numbers such that $a, b, c$ are in AP, $b, c, d$ are GP, $c, d, e$ are in HP. Then $a, c, e$ are in

50% studentsanswered this correctly

Important Questions on Sequences and Series

MEDIUM

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

The sum $\sum_{n = 1}^{\infty} \frac{2 n^{2} + 3 n + 4}{(2 n)!}$ is equal to :

HARD

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

\sum_{n = 0}^{\infty} \frac{n^{3} ((2 n)!) + (2 n - 1) (n!)}{(n!) ((2 n)!)} = a e + \frac{b}{e} + c

where

a, b, c \in ℤ

and

e = \sum_{n = 0}^{\infty} \frac{1}{n!}

Then

a^{2} - b + c

is equal to _______

MEDIUM

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

The sum

1^{2} - 2 . 3^{2} + 3 . 5^{2} - 4 . 7^{2} + 5 . 9^{2} - \dots . . + 15 . 29^{2}

is _____ .

MEDIUM

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

Let

S_{K} = \frac{1 + 2 + . . . + K}{K}

and

\sum_{j = 1}^{n} {S^{2}}_{j} = \frac{n}{A} (B n^{2} + C n + D)

where

A, B, C, D \in N

and

A

Has least value then

EASY

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

For a sequence, if

S_{n} = \frac{5^{n} - 2^{n}}{2^{n}},

then its fourth term is

MEDIUM

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

The sum of the first $20$ terms of the series $5 + 11 + 19 + 29 + 41 + . . .$ is

HARD

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

a_{n} = \frac{- 2}{4 n^{2} - 16 n + 15}

, then

a_{1} + a_{2} + \dots . + a_{25}

is equal to:

HARD

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

For

k \in N,

let

\frac{1}{α (α + 1) (α + 2) \dots \dots . (α + 20)} = \sum_{K = 0}^{20} \frac{A_{k}}{α + k},

where

α > 0 .

Then the value of

100 {(\frac{A_{14} + A_{15}}{A_{13}})}^{2}

is equal to ____________.

MEDIUM

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

The sum of the series

\frac{1}{x + 1} + \frac{2}{x^{2} + 1} + \frac{2^{2}}{x^{4} + 1} + \dots + \frac{2^{100}}{x^{2^{100}} + 1}

when

x = 2

is:

EASY

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

$Let S_{n} = 1 \cdot (n - 1) + 2 \cdot (n - 2) + 3 \cdot (n - 3) + \dots + (n - 1) \cdot 1, n ⩾ 4 .$

$The sum \sum_{n = 4}^{\infty} (\frac{2 S_{n}}{n!} - \frac{1}{(n - 2)!}) is equal to :$

MEDIUM

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

Let $< a_{n} >$ be a sequence such that $a_{1} + a_{2} + . . . + a_{n} = \frac{n^{2} + 3 n}{(n + 1) (n + 2)}$ . If $28 \sum_{k = 1}^{10} \frac{1}{a_{k}} = p_{1} p_{2} p_{3} . . . p_{m}$ , where $p_{1}, p_{2}, . . . p_{m}$ are the first $m$ prime numbers, then $m$ is equal to

MEDIUM

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

\cos (x - y), \cos x

and

\cos (x + y)

are in

HP

, then

\cos x \sec (y / 2)

is equal to

HARD

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

If the sum of the coefficients of all even powers of

x

in the product

(1 + x + x^{2} + \dots + x^{2 n}) (1 - x + x^{2} - x^{3} + \dots + x^{2 n})

61,

then

n

is equal to

MEDIUM

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

The sum to $10$ terms of the series

$\frac{1}{1 + 1^{2} + 1^{4}} + \frac{2}{1 + 2^{2} + 2^{4}} + \frac{3}{1 + 3^{2} + 3^{4}} + \dots$ is :-

HARD

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

Let

α

and

β

be the roots of

x^{2} - x - 1 = 0,

with

α > β .

For all positive integers

n,

define

a_{n} = \frac{α^{n} - β^{n}}{α - β}, n \geq 1

b_{1} = 1

and

b_{n} = a_{n - 1} + a_{n + 1}, n \geq 2 .

Then which of the following options is/are correct?

MEDIUM

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

Suppose

p, q, r

are real numbers such that

q = p (4 - p), r = q (4 - q), p = r (4 - r) .

The maximum possible value of

p + q + r

HARD

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

s i n^{4} α + 4 c o s^{4} β + 2 = 4 \sqrt{2} s i n α c o s β

α, β \in [0, π]

, then

c o s (α + β) - c o s (α - β)

is equal to

MEDIUM

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

\frac{1^{3} + 2^{3} + 3^{3} . . . . . . upto n terms}{1 \cdot 3 + 2 \cdot 5 + 3 \cdot 7 + . . . . upto n terms} = \frac{9}{5}

then the value of

n

EASY

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

Let

x

and

y

be two positive real numbers such that

x + y = 1

. Then the minimum value of

\frac{1}{x} + \frac{1}{y}

is-

MEDIUM

Mathematics>Algebra>Sequences and Series>Relationship Between Arithmetic Mean and Geometric Mean

Let

f : R \to R

be such that for all

x \in R (2^{1 + x} + 2^{1 - x}), f (x)

and

(3^{x} + 3^{- x})

are in A.P., then the minimum value of

f (x)

If a, b, c,d,e be 5 numbers such that a, b, c are in AP, b, c, d are GP, c,d,e are in HP. Then a, c, e are in

Important Questions on Sequences and Series

Learn and Prepare for any exam you want !

If $a, b, c, d, e$ be 5 numbers such that $a, b, c$ are in AP, $b, c, d$ are GP, $c, d, e$ are in HP. Then $a, c, e$ are in