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# If $5,k,11$ are in AP, then the value of $k$ is(A) $6$ (B) $8$ (C) $7$ (D) $9$

Last updated date: 20th Jun 2024
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Hint: The full form of AP is Arithmetic progression. It is a mathematical sequence in which the difference between two consecutive terms is always a constant.
In this progression, for a given series, the terms used are the first term, the common difference between the two terms and nth term. Suppose, ${a_1},{a_2},{a_3},{a_4},......{a_n}$ is an AP, then the common difference‘d’ can be obtained as:
$d = {a_2} - {a_1} = {a_3} - {a_2} = ...... = {a_n} - {a_{n - 1}}$
Where‘d’ is a common difference, it can be positive, negative or zero.

Given that $5,k,11$ are in AP,
AP stands for Arithmetic progression. If these three terms are in Arithmetic progression then the difference between any two consecutive terms should be the same.
First term$= 5$
Second term$= k$
Third term$= 11$
$\therefore$ Second term$-$ First term$=$ Third term$-$ Second term
$\Rightarrow k - 5 = 11 - k$
$\Rightarrow k + k = 11 + 5$
$\Rightarrow 2k = 16$
$\Rightarrow k = \dfrac{{16}}{2}$
$\Rightarrow k = 8$
Hence the value of k is $8$.

Note:
In mathematics, there are three different types of progressions. They are:
Arithmetic Progression (AP)
Geometric Progression (GP)
Harmonic Progression (HP)
Arithmetic Progression (AP) - A sequence of numbers is called an arithmetic progression if the difference between any two consecutive terms is always the same.
Geometric Progression (GP) – A sequence of numbers is called an arithmetic progression if the difference between any two consecutive terms is always the same
Harmonic Progression (HP) – A sequence of numbers is called a harmonic progression if the reciprocal of the terms are in Arithmetic Progression.