The AM of two numbers exceeds their GM by 10 and HM by 16 . Find the numbers.

Arithmetic mean (AM),Geometric mean (GM) Let the two numbers be a and b Given A M = G M + 10 A M - G M = 10 ⇒ A M - 10 = G M . . . . . . (i) also given A M = H M + 16 A M - H M = 16 ⇒ A M - 16 = H M . . . . . . (ii) But Relation between A M , G M and H M is A M × H M = ( G M ) 2 → 3 Substitute equation (i) and (ii) in equation 3 , A M [ A M - 16 ] = ( A M - 10 ) 2 A M 2 - 16 A M = A M 2 - 20 A M + 100 20 A M - 16 AM = 100 4 A M = 100 ⇒ A M = 100 4 ⇒ A M = 25 Substitute A M = 25 in equation (i) A M - 10 = G M 25 - 10 = G M ⇒ G M = 15 Substitute A M = 25 in equation 2 A M - 16 = H M 25 - 16 = H M ⇒ H M = 9 A M = 25 , G M = 15 , H M = 9 A M = a + b 2 25 = a + b 2 ⇒ a + b = 50 → 3 G M = a b 15 = a b ⇒ ( 15 ) 2 = a b ⇒ a b = 225 ⇒ b = 225 a → 4 Substitute 4 in 3 a + 225 a = 50 ⇒ a 2 + 225 = 50 a a 2 - 50 a + 225 = 0 ⇒ a 2 - 45 - 5 a + 225 = 0 a ( a - 45 ) - 5 ( a - 45 ) = 0 ( a - 45 ) ( a - 5 ) = 0 a = 45 , a = 5 a - 45 = 0 ; a - 5 = 0 a = 45 ; a = 5 Put a = 45 in equation 4 b = 225 45 ⇒ b = 5 Put a = 5 in equation 4 b = 225 5 ⇒ b = 45 Numbers are 5 , 45

EASY

Earn 100

Find the harmonic mean between $\frac{a}{(1 - a b)} and \frac{a}{(1 + a b)}$ .

50% studentsanswered this correctly

Important Questions on Binomial Theorem, Sequences and Series

MEDIUM

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

Given that

n

numbers of arithmetic means are inserted between two sets of numbers

a, 2 b

and

2 a, b

where

a, b \in R

. Suppose further that the

m^{th}

means between these sets of numbers are same, then the ratio

a : b

equals

MEDIUM

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

If harmonic mean of

\frac{1}{2}, \frac{1}{2^{2}}, \frac{1}{2^{3}}, \dots \frac{1}{2^{10}}

\frac{λ}{2^{10} - 1}

, then

λ =

MEDIUM

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

The sum of first three terms of an $A P$ is $48$ . If the product of first and second terms exceeds $4$ times the third term by $12$ . Find the $A P$ .

EASY

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

\frac{1}{2}, \frac{1}{4}

and

\frac{1}{6}

are three continuous terms of a series. Find the nature of the series.

MEDIUM

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

H

is the harmonic mean of numbers

1, 2, 2^{2}, 2^{3}, \dots, 2^{n - 1}

, then what is

n / H

equal to?

MEDIUM

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

x + y + z = 15

, when

a, x, y, z, b

are in A.P. and

\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{5}{3}

when

a, x, y, z, b

are in H.P., then the quadratic equation whose roots are

\frac{1}{a}

and

\frac{1}{b}

MEDIUM

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

The sum of three consecutive terms of an $A P$ is $21$ and the sum of the squares of these terms is $165$ . Find these terms

HARD

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

ln (a + c), ln (c - a), ln (a - 2 b + c)

are in A.P., then

HARD

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

Let

A, G, H

A . M ., G . M .

and

H . M .

of three positive real numbers

a, b, c

, respectively such that

G^{2} = A H,

then prove that

a, b, c

are terms of a

G . P .

HARD

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

If the

(m + 1)^{t h}, (n + 1)^{t h} & (r + 1)^{t h}

terms of an AP are in GP

& m, n, r

are in H P, then the ratio of the common difference to the first term of the AP is -

HARD

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

p, q, r

in harmonic progression and

p & r

be different having same sign then the roots of the equation

p x^{2} + q x + r = 0

are

EASY

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

If the

(n + 1) th

term of a harmonic progression is twice the

(3 n + 1) th

term, find the ratio of the first term to the

(n + 1) th

term.

HARD

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

The

AM

of two numbers exceeds their

GM

10

and

HM

16

. Find the numbers.

HARD

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

a_{i} > 0, i = 1, 2, 3, . . ., 50

and

a_{1} + a_{2} + a_{3} + . . . + a_{50} = 50

, then the minimum value of

\frac{1}{a_{1}} + \frac{1}{a_{2}} + \frac{1}{a_{3}} + . . . + \frac{1}{a_{50}}

is equal to.

EASY

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

The harmonic mean of

4, 8, 16

EASY

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

Find the harmonic mean between $7 and 9$

MEDIUM

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

Harmonic mean of the reciprocal of even numbers from

12

190

EASY

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

Cotyledons are also called-

MEDIUM

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

Represent the union of two sets by Venn diagram for each of the following.

$X = {x | x$ is a prime number between $80$ and $100}$

$Y = {y | y$ is an odd number between $90$ and $100}$

EASY

Mathematics>Algebra>Binomial Theorem, Sequences and Series>Finite Sequences

When all the observations are same, then the relation between A.M., G.M., and H.M. is:

Find the harmonic mean between a(1-ab) and a(1+ab).

Important Questions on Binomial Theorem, Sequences and Series

Learn and Prepare for any exam you want !

Find the harmonic mean between $\frac{a}{(1 - a b)} and \frac{a}{(1 + a b)}$ .