 QUESTION

# Is 200 any term of the sequence 3,7,11,15, ........

Hint: To solve this question we will apply the formula of Arithmetic Progression which is given as $a_n$ = a + (n - 1) d, where a is the first term, n is the number of terms, d the common difference and $a_n$ is the last term. And we will assume $a_n$ = 200.

To solve this question, we first of all need to determine what is the first term a, common difference d and the value of the last term.
We are given the sequence as,
3,7,11,15, ........
This is an A.P.

Observing this sequence we can say that a = first term = 3, the common difference is d = 7 – 3 = 4 and the term which has to check if it belongs to a given sequence is $a_n$ = 200.
To check if the given term belongs to the sequence we will apply the formula of Arithmetic Progression which is given as $a_n$ = a + (n - 1)d, where a is the first term, n is the number of terms, d the common difference and $a_n$ is the last term.
Using the above formula, we have,
$a_n$ = a + (n - 1) d
substituting the values of a, d and an we get,
$\begin{array}{*{35}{l}} {{a}_{n}}~=a+\left( n-1 \right)d \\ \Rightarrow 200=3+\left( n-1 \right)4 \\ \Rightarrow 200=3+4n-4 \\ \Rightarrow 201=4n \\ \Rightarrow n=\dfrac{201}{4} \\ \end{array}$
Therefore, we obtained the value of n as a fraction which is not possible because the value of n should be a natural number.
This implies that we have inserted a wrong term in the sequence.
Therefore, 200 is not the term of the sequence 3,7,11, 15,.....

Note: The point to note in this question can be at the place where we are assuming 200 to be the nth term of the given Arithmetic progression. We need to do so, to check if there are no contradictions till the end of the question, which will be helpful to determine if 200 belongs to the given sequence.