Higher Algebra>Chapter 6 - Harmonical Progression, Theorems Connected with The Progression>EXAMPLES. VI.a.>Q 2

HARD

JEE Main

IMPORTANT

Earn 100

Insert two harmonic means between $5$ and $11 .$

Important Questions on Harmonical Progression, Theorems Connected with The Progression

HARD

JEE Main

IMPORTANT

Higher Algebra>Chapter 6 - Harmonical Progression, Theorems Connected with The Progression>EXAMPLES. VI.a.>Q 3

Insert four harmonic means between

\frac{2}{3}

and

\frac{2}{13}

.

HARD

JEE Main

IMPORTANT

Higher Algebra>Chapter 6 - Harmonical Progression, Theorems Connected with The Progression>EXAMPLES. VI.a.>Q 4

If

12

and

9 \frac{3}{5}

are the geometric and harmonic means between two numbers respectively, find them.

HARD

JEE Main

IMPORTANT

Higher Algebra>Chapter 6 - Harmonical Progression, Theorems Connected with The Progression>EXAMPLES. VI.a.>Q 5

If the ratio of harmonic mean to their geometric mean between two quantities is

12 : 13

. Prove that the quantities are in the ratio of

4 : 9

.

MEDIUM

JEE Main

IMPORTANT

Higher Algebra>Chapter 6 - Harmonical Progression, Theorems Connected with The Progression>EXAMPLES. VI.a.>Q 6

If $a, b, c$ are in $H . P .$ , show that $a : (a - b) = (a + c) : (a - c)$

HARD

JEE Main

IMPORTANT

Higher Algebra>Chapter 6 - Harmonical Progression, Theorems Connected with The Progression>EXAMPLES. VI.a.>Q 7

If

m^{th}

term of an

H . P .

is equal to

n,

and

n^{th}

term be equal to

m

. Prove that

{(m + n)}^{th}

term is equal to

\frac{m n}{m + n}

.

HARD

JEE Main

IMPORTANT

Higher Algebra>Chapter 6 - Harmonical Progression, Theorems Connected with The Progression>EXAMPLES. VI.a.>Q 8

If the $p^{th}, q^{th}, r^{th}$ terms of an $H . P .$ be $a, b, c$ , respectively. Prove that $(q - r) b c + (r - p) c a + (p - q) a b = 0$ .

HARD

JEE Main

IMPORTANT

Higher Algebra>Chapter 6 - Harmonical Progression, Theorems Connected with The Progression>EXAMPLES. VI.a.>Q 9

If $b$ is the harmonic mean between $a$ and $c,$ prove that $\frac{1}{b - a} + \frac{1}{b - c} = \frac{1}{a} + \frac{1}{c}$ .

HARD

JEE Main

IMPORTANT

Higher Algebra>Chapter 6 - Harmonical Progression, Theorems Connected with The Progression>EXAMPLES. VI.a.>Q 10

Find the sum of the

n

terms of the series whose

n^{th}

term is

3 n^{2} - n

.