Question

Is 184 a term in the sequence 3, 7, 11, ….?
(a) Yes
(b) No
(c) Cannot be determined
(d) None of the above

Answer 1

Hint: We solve this problem by using the \[{{n}^{th}}\] term of an arithmetic progression.
The formula for \[{{n}^{th}}\] term of an arithmetic progression is given as
\[{{T}_{n}}=a+\left( n-1 \right)d\]
Where, \[a\] is the first term of A.P and \[d\] is the common difference.
We take the given number 184 as \[{{n}^{th}}\] term of given A.P and check whether the value of \[n\] is integer or not.

Complete step by step answer:
We are given with arithmetic progression as
3, 7, 11, ….
Let us take the first term of the given A.P as
\[\Rightarrow a=3\]
Now, let us find the common difference of given A.P
We know that the common difference of A.P is given as the difference of any two consecutive terms of the A.P
By using the above condition we get the common difference as
\[\begin{align}
  & \Rightarrow d=7-3 \\
 & \Rightarrow d=4 \\
\end{align}\]
Now, we are asked to find whether 184 is the term in the given A.P or not.
Let us assume that the 184 is the \[{{n}^{th}}\] term of the given A.P
Now, let us check whether our assumption is correct or wrong.
We know that the formula for \[{{n}^{th}}\] term of an arithmetic progression is given as
\[{{T}_{n}}=a+\left( n-1 \right)d\]
Where, \[a\] is the first term of A.P and \[d\] is the common difference.
By using the above formula to 184 then we get
\[\begin{align}
  & \Rightarrow 184=3+\left( n-1 \right)4 \\
 & \Rightarrow \left( n-1 \right)\times 4=181 \\
 & \Rightarrow n=\dfrac{181}{4}+1 \\
 & \Rightarrow n=\dfrac{185}{4} \\
\end{align}\]
Here, we can see that the value of \[n\] is not an integer.
We know that the number of a term will never be a fraction.
Therefore we can conclude that 184 is not a term in the given A.P.

So, the correct answer is “Option b”.

Note: We can solve this problem in other methods also.
We are given that the arithmetic progression as
3, 7, 11, ….
Now, let us find the common difference of he given A.P
We know that the common difference of A.P is given as the difference of any two consecutive terms of the A.P
By using the above condition we get the common difference as
\[\begin{align}
  & \Rightarrow d=7-3 \\
 & \Rightarrow d=4 \\
\end{align}\]
By using the above common difference let us extend some of the terms of the given A.P then we get
3, 7, 11, 15, 19, 23, 27, 31, …
Here we can see that all the numbers are odd numbers only.
The common difference is an even number and the terms of given A.P are odd numbers.
So, we can say that the given A.P includes only odd numbers.
We are asked to check the number 184 which is an even number.
Therefore we can conclude that 184 is not a term in the given A.P
So, option (b) is the correct answer.