Let a sequence of numbers be denoted as t 1 , t 2 , t 3 , … , where t 1 = 1 and t n = t n - 1 + n . ( n is a natural number). Find t 2 , t 3 , t 4 , t 10 , t 2017 .

Given t 1 = 1 and t n = t n - 1 + n , n ∈ N . Put, n = 2 , to get t 2 = t 1 + 2 = 1 + 2 = 3 . Now, put n = 3 , to get t 3 = t 2 + 2 ⇒ t 3 = 1 + 2 + 3 = 3 + 3 = 6 . Now, put n = 4 , to get t 4 = t 3 + 2 ⇒ t 4 = 1 + 2 + 3 + 4 = 6 + 4 = 10 . In the same way, we can write t 10 = 1 + 2 + … + 10 = 55 and t 2017 = 1 + 2 + 3 + . . . + 2017 Using the formula for the sum of first n natural numbers i.e. 1 + 2 + 3 + . . . + n = n n + 1 2 , we get t 2017 = 2017 × 2018 2 ⇒ t 2017 = 2017 × 1009 = 2035153 . ∴ t 2 = 3 , t 3 = 6 , t 4 = 10 , t 10 = 55 and t 2017 = 2035153 .

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Define a function on the positive integers recursively by $f (1) = 2, f (n) = f (n - 1) + 1$ $i f n$ is even, and $f (n) = f (n - 2) + 2$ if $n$ is odd and greater than $1 .$ What is $f (2017) - 2000 ?$

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Important Questions on Sequence and Series

EASY

Mathematics>Algebra>Sequence and Series>Recurrence Relation

Let

x_{1}

be a positive real number and for every integer

n \geq 1

let

x_{n + 1} = 1 + x_{1} x_{2} \dots x_{n - 1} x_{n} .

x_{5} = 43,

what is the sum of digits of the largest prime factor of

x_{6}

EASY

Mathematics>Algebra>Sequence and Series>Recurrence Relation

Are two consecutive Fibonacci numbers relatively prime?

MEDIUM

Mathematics>Algebra>Sequence and Series>Recurrence Relation

The sequence

S_{1}, S_{2}, S_{3}, \dots \dots S_{10}

has the property that every term beginning with the third is the sum of the previous two. That is,

S_{n} = S_{n - 2} + S_{n - 1}

for

n \geq 3

. Suppose that

S_{9} = 110

and

S_{7} = 42

. What is

S_{4} ?

HARD

Mathematics>Algebra>Sequence and Series>Recurrence Relation

The increasing sequence of positive integers

a_{1}, a_{2}, a_{3},

..... has the property that

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

for all

n \geq 1 .

Suppose that

a_{7} = 120 .

What is

a_{8}

HARD

Mathematics>Algebra>Sequence and Series>Recurrence Relation

A sequence

\{a_{n}\}

is defined by

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

. Find the value of

(1100 + 2008 a_{2007})

MEDIUM

Mathematics>Algebra>Sequence and Series>Recurrence Relation

Let

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

be a sequence for which

a_{1} = 2, a_{2} = 3

and

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

for each positive integer

n \geq 3

. What is

a_{2006} ?

MEDIUM

Mathematics>Algebra>Sequence and Series>Recurrence Relation

Let a sequence of numbers be denoted as

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

where

t_{1} = 1

and

t_{n} = t_{n - 1} + n .

(

n

is a natural number). Find

t_{2}, t_{3}, t_{4}, t_{10}, t_{2017} .

EASY

Mathematics>Algebra>Sequence and Series>Recurrence Relation

The Fibonacci sequence is defined by

a_{1} = 1 = a_{2}, a_{n} = a_{n - 1} + a_{n - 2}

for

n > 2

. Find

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

, for

n = 1, 2, 3, 4, 5 .

EASY

Mathematics>Algebra>Sequence and Series>Recurrence Relation

If the

11^{th}

term of the Fibonacci sequence is

89

and

13^{th}

term is

233

then, what is the

12^{th}

term?

MEDIUM

Mathematics>Algebra>Sequence and Series>Recurrence Relation

Define a sequence of real numbers

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

a_{1} = 1

and

a_{n + 1}^{3} = 99 a_{n}^{3}

for all

n \geq 1

. Then

{(a_{100})}^{\frac{1}{33}}

equals

MEDIUM

Mathematics>Algebra>Sequence and Series>Recurrence Relation

Define a sequence recursively by

F_{0} = 0, F_{1} = 1

and

F_{n} =

the remainder when

F_{n - 1} + F_{n - 2}

is divided by

3,

for all

n \geq 2

. Thus the sequence starts

0, 1, 1, 2, 0, 2 \dots . .

What is

F_{2017} + F_{2018} + F_{2019} + F_{2020} + F_{2021} + F_{2022} + F_{2023} + F_{2024} ?

HARD

Mathematics>Algebra>Sequence and Series>Recurrence Relation

Two non-decreasing sequences of non-negative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is

N .

The smallest possible value of

N

n .

What is half of

n ?

HARD

Mathematics>Algebra>Sequence and Series>Recurrence Relation

Let

\{a_{k}\}

be a sequence of integers such that

a_{1} = 1

and

a_{m + n} = a_{m} + a_{n} + m n,

for all positive integers

m

and

n .

Then

a_{12}

MEDIUM

Mathematics>Algebra>Sequence and Series>Recurrence Relation

Consider the Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \dots$ Observe and complete the following table by understanding the number pattern followed. After filling the table discuss the pattern followed in addition and subtraction of the numbers of the sequence.

Steps	Pattern $1$	Pattern $2$
$i)$	$1 + 3 = 4$	$5 - 1 = 4$
$ii)$	$1 + 3 + 8 =$	?
$iii)$	$1 + 3 + 8 + 21 =$	?
$iv)$	?	?

MEDIUM

Mathematics>Algebra>Sequence and Series>Recurrence Relation

A function

f

is defined recursively by

f (1) = f (2) = 1

and

f (n) = f (n - 1) - f (n - 2) + n

for all integers

n \geq 3

. What is the value of

[f (2018) - 2000] ?

MEDIUM

Mathematics>Algebra>Sequence and Series>Recurrence Relation

Suppose that

(u_{n})

is a sequence of real numbers satisfying

u_{n + 2} = 2 u_{n + 1} + u_{n},

and that

u_{3} = 9

and

u_{6} = 128 .

What is

u_{5} ?

HARD

Mathematics>Algebra>Sequence and Series>Recurrence Relation

Let

a_{n}

denote the number of all

n

digit positive integers formed by the digits

0, 1

or both such that no consecutive digits in them are

0 .

Let

b_{n} =

the number of such

n

digit integers ending with digit

1

and

c_{n} =

the number of such

n

digit integers ending with digit

0 .

The value of

b_{6}

EASY

Mathematics>Algebra>Sequence and Series>Recurrence Relation

Find the

10^{th}

term of the Fibonacci sequence.

MEDIUM

Mathematics>Algebra>Sequence and Series>Recurrence Relation

Let $a_{1}, a_{2}, . . .,$ be a sequence with the following properties.

$I .$ $a_{1} = 1,$ and

$II .$ $a_{2 n} = n \cdot a_{n}$ for any positive integer $n .$

Find the sum of digits of $' n'$ if $a_{2^{100}} = 2^{n} .$

HARD

Mathematics>Algebra>Sequence and Series>Recurrence Relation

Suppose that the sequence $\{a_{n}\}$ is defined by

$a_{1} = 2,$ and $a_{n + 1} = a_{n} + 2 n,$ when $n \geq 1$ .

What is $a_{100}$ ?

Define a function on the positive integers recursively by f(1)=2 , f(n)=f(n-1)+1 if n is even, and f(n)=f(n-2)+2 if n is odd and greater than 1. What is f(2017)-2000?

Important Questions on Sequence and Series

Learn and Prepare for any exam you want !

Define a function on the positive integers recursively by $f (1) = 2, f (n) = f (n - 1) + 1$ $i f n$ is even, and $f (n) = f (n - 2) + 2$ if $n$ is odd and greater than $1 .$ What is $f (2017) - 2000 ?$