Question

In the A.P. 5, 7, 9, 11, 13,………………… the sixth term which is prime is?
a) 15
b) 19
c) 17
d) 23

Answer 1

Hint: A.P stands for Arithmetic Progression, and defined as a sequence of terms with equal successive differences. Here, successive means difference of second to first and third to second etc. Which is known as the common difference of an A.P. Prime number is a number which has only two factors that are 1 and number itself.

Complete step-by-step answer:
Given sequence of the A.P is
5, 7, 9, 11, 13,…………………(i)
So, we need to determine the sixth term which is prime. Now, as we know A.P stands for arithmetic progression which has the same successive difference between the terms (second – first = third – second) which is termed as the common difference of the A.P.
Now, observe the given sequence of the A.P in the equation (i) and we can get common difference of the sequence as
c.d. = second term – first term = 7 – 5 = 2
c.d. = third term – second term = 9 – 7 = 2
c.d. = fourth term – third term = 11 – 9 = 2
Where, c.d. stands for common difference. Hence, we get the common difference of the given sequence is 2. So, we can write the next term (not given) of the sequence by adding ‘2’ to its previous number.
So, we have 5 terms in the given sequence. So, the 6th term of the sequence can be given as
Sixth term = 13 + 2 = 15
Seventh term = 15 + 2 = 17
Eight term = 17 + 2 = 19
Similarly, we can write the sequence as
5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29,………………….(ii)
Now, as we need to determine the sixth term which is prime, it means we need to calculate the terms from the sequence (ii) which are prime numbers. As we know, the prime number is a number which has only two factors that are ‘1’ and the number itself. So, we can get sequence of prime numbers from the sequence of equation (ii) as
5, 7, 9, 11, 13, 17, 19, 23, 29,………………….
Hence the sixth term which is prime is 19.

So, the correct answer is “Option b”.

Note: One may go wrong if he or she gives the answer of the problem as the 6th term of the given sequence i.e. 15. So, please take care that we need to find the 6th term which is prime.
One may think that there may be a direct formula to get the sixth prime number which is wrong. There is no direct approach to get the prime numbers of the sequence. So, just write the whole sequence up to which you do not get the required prime number. Yes, there is a direct identity to get the nth term of the sequence, but it will not help to solve this problem. Identity to get nth term of an A.P is given as
${{T}_{n}}=a+\left( n-1 \right)d$
Where a, d are the first term and common difference of the A.P and ${{T}_{n}}$ is the nth term of the sequence.