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

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Last updated date: 13th Jun 2024
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Answer
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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 \[d\].

Complete step by step answer:
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.