(A) Given that $a, A_1, A_2, b$ are in $A.P.$The sum of $A.M.'s$ inserted between $a$ and $b$ is given by $A_1 + A_2 = n \times \frac{a+b}{2}$,where $

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(A) Given that $a, A_1, A_2, b$ are in $A.P.$The sum of $A.M.'s$ inserted between $a$ and $b$ is given by $A_1 + A_2 = n \times \frac{a+b}{2}$,where $n=2$.So,$A_1 + A_2 = 2 \times \frac{a+b}{2} = a + b$ .....$(i)$Given that $a, G_1, G_2, b$ are in $G.P.$The product of $G.M.'s$ inserted between $a$ and $b$ is given by $G_1 G_2 = (ab)^{n/2}$,where $n=2$.So,$G_1 G_2 = (ab)^{2/2} = ab$ .....$(ii)$Therefore,$\frac{A_1 + A_2}{G_1 G_2} = \frac{a + b}{ab}$.

Class 11 Mathematics Sequences and Series Arithmetic progression

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If $A_1, A_2$ are the two $A.M.'s$ between two numbers $a$ and $b$ and $G_1, G_2$ are two $G.M.'s$ between the same two numbers,then $\frac{A_1 + A_2}{G_1 G_2} = $

A
$\frac{a + b}{ab}$
B
$\frac{a + b}{2ab}$
C
$\frac{2ab}{a + b}$
D
$\frac{ab}{a + b}$

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Class 11

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Sequences and Series

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Arithmetic progression

If the sum of $n$ terms of an $AP$ is given by $S_{n} = n^{2} + n$,then the common difference of the $AP$ is

MediumKCET 2020

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The sum of the first four terms of an arithmetic progression is $56$. The sum of the last four terms is $112$. If its first term is $11$,then the number of terms is:

MediumWBJEE 2025

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The arithmetic mean of the nine numbers in the given set $\{9, 99, 999, \dots, 999999999\}$ is a $9$-digit number $N$,all whose digits are distinct. The number $N$ does not contain the digit:

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Let the sum of the first $n$ terms of a non-constant $A.P.$,$a_1, a_2, a_3, \dots$ be $S_n = 50n + \frac{n(n - 7)}{2}A$,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

DifficultJEE MAIN 2019

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If the $p^{th}$ term of an arithmetic progression is $q$ and its $q^{th}$ term is $p$,then what is its $(p + q)^{th}$ term?

Difficult

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If $A_1, A_2$ are the two $A.M.'s$ between two numbers $a$ and $b$ and $G_1, G_2$ are two $G.M.'s$ between the same two numbers,then $\frac{A_1 + A_2}{G_1 G_2} = $

Similar Questions

If the sum of $n$ terms of an $AP$ is given by $S_{n} = n^{2} + n$,then the common difference of the $AP$ is

The sum of the first four terms of an arithmetic progression is $56$. The sum of the last four terms is $112$. If its first term is $11$,then the number of terms is:

The arithmetic mean of the nine numbers in the given set $\{9, 99, 999, \dots, 999999999\}$ is a $9$-digit number $N$,all whose digits are distinct. The number $N$ does not contain the digit:

Let the sum of the first $n$ terms of a non-constant $A.P.$,$a_1, a_2, a_3, \dots$ be $S_n = 50n + \frac{n(n - 7)}{2}A$,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

If the $p^{th}$ term of an arithmetic progression is $q$ and its $q^{th}$ term is $p$,then what is its $(p + q)^{th}$ term?

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