 Hello and welcome to the session. Let us understand the following question today. Show that A1, A2, An form an AP where An is defined as below. An is equal to 3 plus 4n. Also find the sum of the first 15 terms. Now, before writing the solution let us understand an AP. A1, A2, A3 so on forms an AP if 2 minus A1 is equal to D. A3 minus A2 is equal to D and so on. That is the difference of 2 terms. A term and a preceding term is always constant that is our D. So, this is our key idea to the question. Now let us write the solution. Given to us is A is equal to 3 plus 4n. So, therefore A1 is equal to 3 plus 4 into 1 which is equal to 7. Similarly, A2 is equal to 3 plus 4 into 2 which is equal to 11 and A3 is equal to 3 plus 4 into 3 which is equal to 15 and A4 is equal to 3 plus 4 into 4 which is equal to 19. Now, it implies A2 minus A1 is equal to 11 minus 7. Similarly, we will find A3 minus A2 which is equal to 15 minus 11. A4 minus A3 is equal to 19 minus 15. Now, we see that A2 minus A1 is equal to 4, A3 minus A2 is equal to 4 and similarly A4 minus A3 is equal to 4. Here we see that we are getting all the fours. So, which is our D? So, which is constant? Therefore, D is constant and equal to 4. Thus, the common difference that is D of a term and the preceding term is a constant. Hence, A1 A2 A3 when A n is equal to 3 plus 4n is an AP. Hence, A1 A2 A3 so on is an AP. Now, let's find the sum. Now, from above we can see our AP will be as 7, 11, 15, 19 and so on. So, the AP form is 7, 11, 15, 19 and so on. Now, here A is equal to 7, D is equal to 11 minus 7 which is equal to 4 and since we have to find the sum of first 15 terms so, n is equal to 15. Now, let us find the sum. Sn is equal to n by 2 multiplied by 2A plus n minus 1D. Now, we have to find S15. So, S15 is equal to 15 by 2 multiplied by 2 into 7 plus 15 minus 1 multiplied by 4 which is equal to 15 by 2 multiplied by 14 plus 14 into 4 which is equal to 15 by 2 multiplied by 14 plus 56 which is equal to 15 by 2 multiplied by 70. Now, this gets cancelled by 35. So, it is equal to 15 into 35 which is equal to 525. So, the required sum is 525. I hope you understood the question. Bye and have a nice day.