 Hi and welcome to the session I am Shashi and I am going to help you to solve the following question. Question is, if the third and the ninth terms of an AP are 4 and minus 8 respectively, which term of this AP is 0? Let us start with the solution, nth term of AP we know is given by a n is equal to a plus n minus 1 multiplied by d, where a is the first term of AP and d is the common difference between the two consecutive terms of the AP. Now we know third term of AP given in the question is 4, so we can write a3 is equal to 4, we also know that a3 is given by a plus 3 minus 1 multiplied by d. Now third term of the AP is equal to 4, so we can substitute the value of a3 and get 4 is equal to a plus 2d, where a is the first term of the AP and d is the common difference. Now we are given ninth term of the AP is minus 8, so we can write a9 is equal to minus 8, now this is also equal to a plus 9 minus 1 multiplied by d, where a is the first term of AP and d is the common difference. Now ninth term is we know is equal to minus 8, so we will substitute its value here, so we get minus 8 is equal to a plus 8d. Now let us name this equation as equation 1 and this equation as equation 2, now subtracting equation 2 from equation 1 we get 4 minus minus 8 is equal to a plus 2d minus a plus 8d, now this implies 12 is equal to a plus 2d minus a minus 8d and a will get cancelled, this implies 12 is equal to minus 60 or we can write minus 60 is equal to 12, this implies d is equal to 12 divided by minus 6 equal to minus 2, so we get d is equal to minus 2. Now substituting d is equal to minus 2 in equation 1 we get a plus 2 multiplied by minus 2 is equal to 4, this implies a minus 4 is equal to 4, this implies a is equal to 4 plus 4, this further implies a is equal to 8, therefore we get a is equal to 8. Now the common difference of the a p is equal to minus 2 and the first term of the a p is equal to 8, now let us assume the nth term of the a p is equal to 0, we have to find which term of the a p is equal to 0, so we are assuming let nth term of the a p be equal to 0. Now we know nth term is equal to a plus n minus 1 multiplied by d where a is the first term of a p and d is the common difference, now we know nth term is equal to 0, so we can substitute its value here, a is equal to 8, so we will substitute its value here and d is equal to minus 2, so we write minus 2 in place of d, now this implies minus 8 is equal to minus 2 multiplied by n minus 1, this implies minus 8 divided by minus 2 is equal to n minus 1, now minus n minus time will get cancelled and we will get to 4 the 8, now 4 is equal to n minus 1 or we can write n minus 1 is equal to 4, this implies n is equal to 4 plus 1, this further implies n is equal to 5, so we get 5th term of the a p is equal to 0, so our required answer is 5th term, this completes the session hope you understood the session take care and goodbye.