If one $AM 'A'$ and two $GM$ $p$ and $q$ are inserted between two given numbers, then find the value of $\frac{p^{2}}{q}+\frac{q^{2}}{p}$
- $A$
- $2A$
- $3A$
- $4A$
The Correct Option is B
Solution and Explanation
Then, $A=\frac{a+b}{2} \ldots(i)$
Also, $a, p, q, b$ are in $G P$.
$\frac{p}{a}=\frac{q}{a}=\frac{b}{q}$
$\Rightarrow \frac{p^{2}}{q}=a$ and $\frac{q^{2}}{p}=b$
$\therefore \frac{p^{2}}{q}+\frac{q^{2}}{p}=a+b=2 A$ [Using e (i)]
Top Questions on Arithmetic Progression
- Assertion (A): The sum of the first fifteen terms of the AP $21, 18, 15, 12, \dots$ is zero.
Reason (R): The sum of the first $n$ terms of an AP with first term $a$ and common difference $d$ is given by: \[ S_n = \frac{n}{2} \left[ a + (n - 1) d \right]. \]- CBSE Class X - 2024
- Mathematics
- Arithmetic Progression
- The sum of the first three terms of an AP is 30 and the sum of the last three terms is 36. If the first term is 9, then the number of terms is :
- CBSE Class X - 2024
- Mathematics
- Arithmetic Progression
- How many terms of the $A.P.$ $27, 24, 21, . . .$ must be taken so that their sum is 105? Which term of the A.P. is zero?
- CBSE Class X - 2024
- Mathematics
- Arithmetic Progression
- In an A.P., if \(a = 8\) and \(a_{10} = -19\), then value of \(d\) is:
- CBSE Class X - 2024
- Mathematics
- Arithmetic Progression
- Assertion (A): The sum of the first fifteen terms of the AP $21, 18, 15, 12, \dots$ is zero.
Reason (R): The sum of the first $n$ terms of an AP with first term $a$ and common difference $d$ is given by: \[ S_n = \frac{n}{2} \left[ a + (n - 1) d \right]. \]- CBSE Class X - 2024
- Mathematics
- Arithmetic Progression
Questions Asked in VITEEE exam
- Two identical blocks A and B, each of mass 'm' resting on a smooth floor are connected by a light spring of natural length L and spring constant K, with the spring at its natural length. A third identical block 'C' (mass m) moving with a speed v along the line joining A and B collides with A. the maximum compression in the spring is
- VITEEE - 2023
- work, energy and power
- Assertion :
The binding energy of the nucleus increases with the increase in atomic number.
Reason :
Heavier elements have a greater number of non-radioactive isotopes than radioactive isotopes. - The half-life of radioactive radon is 3.8 days. The time at the end of which \(\frac{1}{20}th\) of the Radon sample will remain undecayed is (given log10e=0.4343)
- Which element has more electron gain enthalpy among chalcogen group?
- VITEEE - 2022
- sulphur
- Coin is tossed till heads is obtained what is expectation of no. of coin tosses
- VITEEE - 2022
- Random Variables
Concepts Used:
Arithmetic Progression
Arithmetic Progression (AP) is a mathematical series in which the difference between any two subsequent numbers is a fixed value.
For example, the natural number sequence 1, 2, 3, 4, 5, 6,... is an AP because the difference between two consecutive terms (say 1 and 2) is equal to one (2 -1). Even when dealing with odd and even numbers, the common difference between two consecutive words will be equal to 2.
In simpler words, an arithmetic progression is a collection of integers where each term is resulted by adding a fixed number to the preceding term apart from the first term.
For eg:- 4,6,8,10,12,14,16
We can notice Arithmetic Progression in our day-to-day lives too, for eg:- the number of days in a week, stacking chairs, etc.
Read More: Sum of First N Terms of an AP