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In a sequence $\{a_{n}\}, a_{1} = 2, a_{n + 1} = 1 - \frac{1}{a_{n}}$ for $n \geq 1, n \in N$ . Let $P_{n}$ be the product of its first $n$ terms, then the value of $P_{2023}$ is equal to

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Important Questions on Sequences and Series

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

If the product of n positive numbers is unity, then their sum is

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

The number of terms common to both the arithmetic progressions

2, 5, 8, 11, \dots, 179

and

3, 5, 7, 9, \dots, 101

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

The value of

\frac{3}{4} + \frac{5}{36} + \frac{7}{144} + \dots \dots + \frac{17}{5184} + \frac{19}{8100}

HARD

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

Consider the sequence

a_{1}, a_{2}, a_{3}, \dots \dots

such that

a_{1} = 1, a_{2} = 2

and

a_{n + 2} = \frac{2}{a_{n + 1}} + a_{n}

for

n = 1, 2, 3, \dots

(\frac{a_{1} + \frac{1}{a_{2}}}{a_{3}}) \cdot (\frac{a_{2} + \frac{1}{a_{3}}}{a_{4}}) \cdot (\frac{a_{3} + \frac{1}{a_{4}}}{a_{5}}) \dots (\frac{a_{30} + \frac{1}{a_{31}}}{a_{32}}) = 2^{α} ({}^{61}C_{31})

then

α

is equal to

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

The sum of the following series

1 + 6 + \frac{9 (1^{2} + 2^{2} + 3^{2})}{7} + \frac{12 (1^{2} + 2^{2} + 3^{2} + 4^{2})}{9} + \frac{15 (1^{2} + 2^{2} + \dots + 5^{2})}{11} + . . . .

up to

15

terms, is:

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

Find odd one out;

1, 5, 14, 30, 51, 55, 91

HARD

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

{(1 + x - 2 x^{2})}^{6} = A_{0} + \sum_{r = 1}^{12} A_{r} x^{r}

, then value of

A_{2} + A_{4} + A_{6} + \dots + A_{12}

EASY

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

Find the missing (?) in the series- $14, 28, 42, 56, ?, 84, 98$

1. $68$
2. $70$
3. $72$
4. $74$

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

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:

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

0 < x < 1

and

y = \frac{1}{2} x^{2} + \frac{2}{3} x^{3} + \frac{3}{4} x^{4} + \dots \dots

, then the value of

e^{1 + y}

x = \frac{1}{2}

is:

HARD

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

\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 _______

EASY

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

Select the correct set of symbols:

21 9 13 7 = 195

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

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>Introduction to Sequences and Series

On simplification the product

$(x_{1} - y_{1}) (x_{2} + y_{2}) . . . (x_{10} + y_{10})$

How many such terms are there which will have only single $x$ and rest $y$ 's ?

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

A sequence is given in which one term (or more terms) is missing. From the given options, choose the correct one that can complete the sequence.

$1, 6, 15, ?, 45, 66, 91$

EASY

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

Complete the following series each of which follows a certain pattern:

$3, 12, 27, 48, 75, 108,_____?$

EASY

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

Find the missing term in the following series:

$2, 3, 4, 6, 8, 12, 16, 24, ?$

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

A car driver increases the average speed of his car by

3 km / h

every hour. The total distance travelled in

7 h

if the distance covered in the first hour was

30 km

, is:

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

The value of

\sqrt{7 + \sqrt{7 - \sqrt{7 + \sqrt{7 - \dots \infty}}}}

MEDIUM

Mathematics>Algebra>Sequences and Series>Introduction to Sequences and Series

Let

\frac{1}{x_{1}}, \frac{1}{x_{2}}, \dots, \frac{1}{x_{n}} (x_{i} \neq 0

for

i = 1, 2, \dots ., n)

be in A.P. such that

x_{1} = 4

and

x_{21} = 20

.

n

is the least positive integer for which

x_{n} > 50

, then

\sum_{i = 1}^{n} (\frac{1}{x_{i}})

is equal to

In a sequence an,a1=2,an+1=1-1an for n≥1,n∈N. Let Pn be the product of its first n terms, then the value of P2023 is equal to

Important Questions on Sequences and Series

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In a sequence $\{a_{n}\}, a_{1} = 2, a_{n + 1} = 1 - \frac{1}{a_{n}}$ for $n \geq 1, n \in N$ . Let $P_{n}$ be the product of its first $n$ terms, then the value of $P_{2023}$ is equal to