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What is the value of $d$ for the given condition? $a = 3.5;{\text{ }}n = 105;{\text{ }}{a_n} = 3.5$

Last updated date: 13th Jun 2024
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Hint: In this problem we have to find the value of common difference $d$ by using given conditions. They gave the initial value $a$, total number of elements $n$ and ${n^{th}}$ term of the arithmetic progression. By using the relations in the arithmetic progression we are going to solve this problem.

Formula used:
We know that, nth term of an arithmetic progression with the initial term as $a$ and $d$ be the common difference, is
${a_n} = a + (n - 1)d$
Here, the given information is a series of an arithmetic progression. So, we will apply the values in the above formula.
Then we can find the value for $d$.

It is given that; $a = 3.5;\,{\text{ }}n = 105;{\text{ }}{a_n} = 3.5$
It means the series is in arithmetic progression. So, the initial term is $a = 3.5$, the value of n is $105$ and the nth term is ${a_n} = 3.5$.
We have to find the value of the common difference that is $d$.
We know that, nth term of an arithmetic progression with the initial term as $a$ and $d$ be the common difference, is
${a_n} = a + (n - 1)d$
Now, substitute the value in the above formula we get,
$a = 3.5;\,{\text{ }}n = 105;{\text{ }}{a_n} = 3.5$
$\Rightarrow {a_{105}} = 3.5 + (105 - 1)d$
Let us substituting for ${a_{105}} = 3.5$ and Simplifying we get,
$\Rightarrow 3.5 = 3.5 + (105 - 1)d$
Simplifying we get,
$\Rightarrow d = 0$

$\therefore$ The value of $d$ is $0$.

Note:
A progression is a special type of sequence for which it is possible to obtain a formula for the nth term. The Arithmetic Progression is the most commonly used sequence in maths with easy to understand formulas. Let’s have a look at its three different types of definitions.
Definition 1: A mathematical sequence in which the difference between two consecutive terms is always a constant and it is abbreviated as AP.
Definition 2: An arithmetic sequence or progression is defined as a sequence of numbers in which for every pair of consecutive terms, the second number is obtained by adding a fixed number to the first one.