Question

12. Find the sixth prime number on an AP 5, 7, 9, 11, ...​

Answer 1

Answer:

5, 7, 9, 11, 13, 17, 19, 23, 29,…………………. Hence the sixth term which is prime is 19.

Answer 2

$\Large{\underbrace{\underline{\sf{Understanding\: the\; Question}}}}$

Here in this question, concept of arithmetic progression (AP) is used. We know that the AP is a series of number which has same common difference. We are given first 4 terms of the particular term and we are asked to find sixth prime number of the AP. We have to find some next terms of the AP so that we can find prime numbers in AP.

So let's start!!

Given AP:

5,7,9,11...

From here:-

• First term, A=5

• Common difference, d= a2-a1=7-5=2

Note→ Common difference can be find by subtracting either of two consecutive terms of AP.

Now we have formula for nth term of AP:

★ An=A+(n-1)d

We can see that 3 terms of the AP are prime number (i.e. 5,7 and 11) we have to find 3 more prime number.

Let's find 5th term!

If we are finding 5th term then value of n will be 5.

Now put the the given values in formula!

⇒ An=A+(n-1)d

⇒ A5=5+(5-1)2

⇒ A5=5+(4)2

⇒ A5=5+8

⇒ A5=13

It means 5th term of AP is also prime.

Now find 6th term of AP.

For 6th term n will be 6

⇒ An=A+(n-1)d

Put given values!

⇒ A6= 5+(6-1)2

⇒ A6=5+(5)2

⇒ A6=5+10

⇒A6=15

Here 6th term of AP is not prime number.

So let's find more terms!

Now we are finding 7th term

For 7th term n will be 7

⇒ An=A+(n-1)d

Put given values

⇒ A7=5+(7-1)2

⇒ A7=5+(6)2

⇒ A7=5+12

⇒ A7=17

It is a prime number.

Now we need to find one more prime number.

So let's try with 8th term!

For 8th term, n will be 8.

⇒ An=A+(n-1)d

Put given values

⇒ A8=5+(8-1)2

⇒ A8=5+(7)2

⇒ A8=5+14

⇒ A8=19

Here it is a prime number and is also 6th prime number of AP.

So the required 6th prime number of AP is 19.

★ Additional information:-

More formulae for AP:

To find sum of AP:-

$\bullet\quad\sf S_n=\dfrac{n}{2}(2a+(n-1)d)$

$\bullet\quad\sf S_n=\dfrac{n}{2}(a+an)$

[Here an is the last term.]