Question

Is 402 a term of the sequence: $8,13,18,23,....?$

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 the first term and use the formula ${a_n} = a + (n - 1)d$ by keeping ${a_n} = 402$ and check whether n is a natural number or not. If it is, then 402 is a part of the given sequence.

Formula used: ${a_n} = a + (n - 1)d$, where ${a_n}$= nth term, $a$= first term, $n$= the number of the nth term, $d$= common difference.

Complete step-by-step answer:
We are given a sequence in this question. By simply observing, we can tell that the sequence forms an A.P as there is a common difference between the terms.
First term (a) = $8$
Common difference (d) = $13 - 8 = 5$
Now we will take ${a_n} = 402$ and put all the terms in the general formula of A.P $\Rightarrow {a_n} = a + (n - 1)d$.
$\Rightarrow 402 = 8 + (n - 1)5$
Solving for n,
$\Rightarrow 402 - 8 = (n - 1)5$
Shifting and simplifying,
$\Rightarrow \dfrac{{394}}{5} + 1 = n$
$\Rightarrow 78.8 + 1 = n$
$\Rightarrow 79.8 = n$
From above, we can see that $n = 79.8$. $'n'$ 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 $402$ is not a term of the sequence.

402 is not a term in the sequence.

Note: In order to identify whether a given sequence is an Arithmetic progression or geometric progression, start by subtracting a term from its next term. Do this twice and thrice. If the difference is always the same, then the sequence is an arithmetic progression. Otherwise, divide a term by its previous term twice or thrice. If the ratio is the same, the sequence is geometric progression.