 Hello and welcome to the session. Let us understand the following question today. Show that a1, a2, a3 serve on form and AP where an is defined as below. An is equal to 9 minus 5n. Also find the sum of first 15 terms. Now, let us write the solution. Given to us is, an is equal to 9 minus 5n. So therefore, a1 is equal to 9 minus 5 into 1 which is equal to 4. An a2 is equal to 9 minus 5 into 2 which is equal to minus 1. Similarly, a3 is equal to 9 minus 5 into 3 which is equal to minus 6. An a4 is equal to 9 minus 5 into 4 which is equal to minus 11. Now, which implies a2 minus a1 is equal to minus 1 minus 4 which is equal to minus 5. Similarly, which implies a3 minus a2 is equal to minus 6 minus of minus 1 which is equal to minus 5. Now, which implies a4 minus a3 is equal to minus 11 minus minus 6 which is equal to minus 5. Now, here we see that and so on. Now, all the common difference, this is our D which is coming as constant. So, common difference is constant. Hence, we find that the common difference of a term and the preceding term is constant. So, a1, a2, a3 so on when an is given to us as 5, 9 minus 5 n is an a p. Therefore, a1, a2, a3 so on is an a p. Now, let's finally sum. Now, here we can see our a p formed is 4 minus 1 minus 6 minus 11 and so on. So, the a p formed is 4 minus 1 minus 6 minus 11 and so on. Here, a is equal to 4, d is equal to minus 1 minus 4 which is equal to minus 5. And since we have to find the sum of first 15 terms, so n is equal to 15. Now, we know Sn is equal to n by 2 multiplied by 2 a plus n minus 1 d. And we have to find S15 which is equal to 15 by 2 multiplied by 2 into 4 plus 15 minus 1 multiplied by minus 5. Now, solving this we get which is equal to 15 by 2 multiplied by 8 plus 14 into minus 5 which is equal to 15 by 2 multiplied by 8 minus 70 which is equal to 15 by 2 multiplied by minus 62. Now, this gets cancelled by minus 31 which is equal to minus 465. Hence, S15 is equal to minus 465 which is our required answer. I hope you understood the question. Bye and have a nice day.