Question

Find the common difference of an Arithmetic sequence with the first term and ${105^{th}}$ term equal to $3.5$.
A) \[d = 1\]
B) $d = 0$
C) $d = 0.5$
D) $d = - 1$

Answer 1

Hint:Using the equation for the ${n^{th}}$ term of an arithmetic sequence we can find the common difference of the sequence. All we need is to write the equation, substitute for the known values and solve for the unknown value.

Formula used:For an arithmetic sequence with first term $a$ and common difference $d$, the ${n^{th}}$ term is given by ${a_n} = a + (n - 1)d$.

Complete step-by-step answer:
Given that an Arithmetic sequence has a first term and ${105^{th}}$ term equal to $3.5$.
Let the first term, common difference, ${n^{th}}$ term be $a,d,{a_n}$ respectively.
$ \Rightarrow a = 3.5,{a_{105}} = 3.5$
For an arithmetic sequence with first term $a$ and common difference $d$, the ${n^{th}}$ term is given by,
$ \Rightarrow {a_n} = a + (n - 1)d$
Using this and substituting for $n$ we get,
$ \Rightarrow {a_{105}} = a + (105 - 1)d$
Substituting for $a,{a_{105}}$ we have,
$ \Rightarrow 3.5 = 3.5 + (105 - 1)d$
Subtracting $3.5$ from both sides we get,
$ \Rightarrow 3.5 - 3.5 = 3.5 + (105 - 1)d - 3.5$
$ \Rightarrow 0 = 104d$
The product of two numbers equals zero means either number is zero.
Here, $104 \ne 0 \Rightarrow d = 0$$$$$
Since $d$ represent the common difference,
we have the common difference of the given sequence is $0$.

So, the correct answer is “Option B”.

Additional Information:A sequence with zero common difference means that every term is the same.Here every term is equal to the first term which is equal to $3.5$.
Therefore, the sequence is $3.5,3.5,3.5,$

Note:If we are given first term and common difference, we can find any term of a sequence. Also if we are given any two terms, we can solve for the common difference using them. In every case it is important to express the given data in equation correctly. The common difference can be negative value as well. In that case we get a decreasing sequence.