Question

Is $184$ a term of the sequence $3,7,11,...?$

Answer 1

In this question, we are given a sequence and we have been asked to find whether the given number is a term of the sequence or not. First, observe whether the sequence is forming an A.P or G.P or if there is any other pattern. After observing, you will notice that the sequence is forming an A.P. Note the common difference and first term and use the formula ${a_n} = a + (n - 1)d$ by keeping ${a_n} = 184$ and check whether n is a natural number or not. If it is, then 184 is a part of the given sequence.

Complete step-by-step answer:
We are given a sequence in this question. By simple observation we can tell that the sequence forms an A.P as there is a common difference between the terms.
First term (a) = $3$
Common difference (d) = $7 - 3 = 4$
Now we will take ${a_n} = 184$ and put all the terms in the general formula of A.P $\Rightarrow {a_n} = a + (n - 1)d$
$\Rightarrow 184 = 3 + (n - 1)4$
Solving for n,
$\Rightarrow 184 - 3 = (n - 1)4$
Simplifying,
$\Rightarrow \dfrac{{181}}{4} + 1 = n$
$\Rightarrow 45.25 + 1 = n$
$\Rightarrow 46.25 = n$
From above, we can see that $n = 46.25$. $'n'$ represents the term which can never be in decimals. A term can only be a natural number. But here, n is not a natural number which implies that the number $184$ is not a term of the sequence.

184 is not a term of the series.

Note: 1) The notations in the formula ${a_n} = a + (n - 1)d$ means the following:
${a_n}$= nth term
$a$= first term
$n$= the number of the nth term
$d$= common difference
2) The fact that $'n'$ should only be a natural number should be clearly understood. We can understand this by the following example: In our alphabets, D is the $4^{th}$ alphabet and E is the $5^{th}$ alphabet. There is no $4.5^{th}$ alphabet. Hence, the number of terms should only be natural numbers.