 Hello and welcome to the session. In this session we will discuss a question which says that write a recursive formula for the geometric sequence whose fourth term is minus 12 and fifth term is minus 6. Now before starting the solution of this question we should know some results. First is the nth term of the geometric sequence is given by a m is equal to a 1 into r raised to power n minus 1 where r is the common version and a 1 is the first term of the sequence. And second is the recursive formula for a geometric sequence is given by a m is equal to r into a n minus 1 where n is greater than 1 and the value of first term of the sequence that is a 1 is known. Now these results will work out as a key idea for solving out the given question. Now let us start with the solution of the given question. Now in this question we are given first term, fifth term of the sequence. So we are given that first term of the sequence that is a 4 is equal to minus 12 that is a 5 is equal to minus 6. Now we know that the general term of the geometric sequence is given by a m is equal to a 1 into r raised to power n minus 1 where a 1 is the first term of the sequence n is equal to 4 in this formula and we have a 4 is equal to a 1 into r raised to power 4 minus 1 which implies a 4 is equal to a 1 into r raised to power 3. Now we are given that a 4 is equal to minus 12 so we will put this value here and we have minus 12 is equal to a 1 into r raised to power 3. Let this be equation 1 is equal to 5 in this formula and is equal to a 1 into r raised to power 5 minus 1 if is equal to a 1 into r raised to power 4 also given that a 5 is equal to minus 6 value here we have minus 6 is equal to a 1 into r raised to power 4 let this be equation 2. Now we will divide equation 2 by equation 1 so dividing equation 2 by equation 1 we get a 1 into r raised to power 4 whole upon a 1 into r raised to power 3 is equal to minus 6 upon minus 12 this implies 6 to power 4 minus 3 is equal to now 6 into 2 is 12 so this will be 1 upon 2 this implies 2 raised to power 1 which is equal to r is equal to 1 upon 2 so common ratio is 1 upon 2 now using the result which is given to us in the key idea we know that the recursive formula for a geometric sequence is given by a n is equal to r into a n minus 1 where n is greater than 1 and value of first term of the sequence that is a 1 is known r is equal to 1 upon 2 equals r is equal to 1 upon 2 in this formula n is equal to 1 upon 2 into a n minus 1 where n is greater than 1 the sequence that is a 1 now this is equation number 1 now for finding the value of a 1 then we put into 1 upon 2 in equation 1 and we get minus 12 is equal to a 1 into 1 upon 2 whole cube minus 12 is equal to a 1 into now 1 cube is 1 and 2 cube is h minus 12 is equal to a 1 into 1 upon 8 now we will multiply both sides of this equation by 8 and we get minus 12 into 8 is equal to a 1 into 1 upon 8 into a 1 that is a is equal to minus 9 recursive formula for the minus 1 where n and this completes our session hope you all have enjoyed the session