If a 2 , b 2 , c 2 are in A.P., then show that a b + c , b c + a , c a + b are also in A.P. (when a + b + c ≠ 0 )

a 2 , b 2 , c 2 are in A . P b 2 - a 2 = c 2 - b 2 ( b - a ) ( b + a ) = ( c - b ) ( c + b ) b - a c + b = c - b b + a b - a c + b c + a = c - b b + a c + a b + c - c - a c + b c + a = c + a - a - b b + a c + a 1 c + a - 1 b + c = 1 a + b = 1 c + a ∴ 1 b + c , 1 c + a , 1 a + b are in A.P ⇒ a + b + c b + c , a + b + c c + a , a + b + c a + b are in A.P a b + c + 1 , b c + a + 1 , c a + b + 1 are in A.P a b + c , b c + a , c a + b are in A.P Hence, proved

If a 2 , b 2 , c 2 are in A.P., then show that a b + c , b c + a , c a + b are also in A.P. (when a + b + c ≠ 0 )

a 2 , b 2 , c 2 are in A . P b 2 - a 2 = c 2 - b 2 ( b - a ) ( b + a ) = ( c - b ) ( c + b ) b - a c + b = c - b b + a b - a c + b c + a = c - b b + a c + a b + c - c - a c + b c + a = c + a - a - b b + a c + a 1 c + a - 1 b + c = 1 a + b = 1 c + a ∴ 1 b + c , 1 c + a , 1 a + b are in A.P ⇒ a + b + c b + c , a + b + c c + a , a + b + c a + b are in A.P a b + c + 1 , b c + a + 1 , c a + b + 1 are in A.P a b + c , b c + a , c a + b are in A.P Hence, proved

EASY

Earn 100

If $a^{2}, b^{2}, c^{2}$ are in A.P., then show that $\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}$ are also in A.P. (when $a + b + c \neq 0$ )

Important Questions on Sequences and Series

EASY

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

In an arithmetic progression, if the

k^{t h}

term is

5 k + 1

,

then the sum of first

100

terms is

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

For any three positive real numbers

a, b

and

c

. If

9 (25 a^{2} + b^{2}) + 25 (c^{2} - 3 a c) = 15 b (3 a + c) .

Then

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

Let

a, b, c, d, e

be natural numbers in an arithmetic progression such that

a + b + c + d + e

is the cube of an integer and

b + c + d

is a square of an integer. The least possible value of the number of digits of

c

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

The number of terms in an

A . P .

is even, the sum of the odd terms in it is

24

and that the even terms is

30 .

If the last term exceeds the first term by

10 \frac{1}{2},

then the number of terms in the

A . P .

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

The sum of all two digit positive numbers which when divided by

7

yield

2

5

as remainder is

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

a^{1 / x} = b^{1 / y} = c^{1 / z}

and

a, b, c

are in geometrical progression, then

x, y, z

are in

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

The houses on one side of a road are numbered using consecutive even numbers. The sum of the numbers of all the houses in that row is

170

. If there are at least

6

houses in that row and

a

is the number of the sixth house, then

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

The number of terms common to the two A.P.’s

3, 7, 11, \dots, 407

and

2, 9, 16, \dots, 709

is ____________.

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

Let the sum of the first three terms of an A.P. be

39

and the sum of its last four terms be

178

. If the first term of this A.P. is

10,

then the median of the A.P. is :

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

Three positive numbers form an increasing

G . P .

If the middle term in this

G . P .

is doubled, the new numbers are in

A . P .

Then the common ratio of the

G . P .

is :

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than

200

and less than

220 .

If the second term in it is

12

, then its

4^{th}

term is :

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

Let

X

be the set consisting of the first

2018

terms of the arithmetic progression

1,6, 11, \dots,

and

Y

be the set consisting of the first

2018

terms of the arithmetic progression

9,16,23, \dots

. Then, the number of elements in the set

X \cup Y

is___.

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

Let

a_{1}, a_{2}, a_{3}, \dots \dots, a_{49}

be in

A . P .

such that

Σ_{k = 0}^{12} a_{4 k + 1} = 416

and

a_{9} + a_{43} = 66 .

a_{1}^{2} + a_{2}^{2} + \dots + a_{17}^{2} = 140 m,

then

m

is equal to:

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

If the sum of the first

n

terms of the series

\sqrt{3} + \sqrt{75} + \sqrt{243} + \sqrt{507} + \dots

435 \sqrt{3},

then

n

equals:

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

Let

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

,be in A.P. If

a_{3} + a_{7} + a_{11} + a_{15} = 72

, then the sum of its first

17

terms is equal to :

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

If the sum and product of the first three terms in an

A . P .

are

33

and

1155

, respectively, then a value of its

11^{t h}

term is:

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

Let the sum of the first

n

terms of a non-constant

A . P ., a_{1}, a_{2}, a_{3}, . . . ., a_{n}

50 n + \frac{n (n - 7)}{2} A,

where

A

is a constant. If

d

is the common difference of this

A . P .

, then the ordered pair

(d, a_{50})

is equal to

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

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

MEDIUM

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

x, y, z

are positive numbers in

A . P .

and

\tan^{- 1} x

\tan^{- 1} y

and

\tan^{- 1} z

are also in

A . P .

, then which of the following is correct.

HARD

Mathematics>Algebra>Sequences and Series>Arithmetic Progressions

^{n} C_{4},^{n} C_{5}

and

^{n} C_{6}

are in A.P., then

n

can be

If a2,b2,c2 are in A.P., then show that ab+c,bc+a,ca+b are also in A.P. (when a+b+c≠0 )

Important Questions on Sequences and Series

Learn and Prepare for any exam you want !

If $a^{2}, b^{2}, c^{2}$ are in A.P., then show that $\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}$ are also in A.P. (when $a + b + c \neq 0$ )