 Hello and welcome to the session. In this session we will write geometric sequence with recursive ring and with an explicit formula and translate between the two forms. Now let us find explicit formula for a geometric sequence. Now consider the following sequence here. First term is a1 which is equal to 2. Then second term of the sequence is a2 which is equal to then third term is a3 which is equal to 18. Now let us see ratio. Get rid of the two consecutive terms. Now a2 upon a1 is equal to 6 upon 2 that is equal to 3 then a3 upon a2 is equal to 18 upon 6 which is again 3 and so on then thus t that there is two ratio denoted by r between the two consecutive terms of the given sequence and that common ratio r is equal to 3 thus it is a geometric sequence. Now let us establish a pattern in the sequence which will give us the explicit formula. Now we have the first term a1 is equal to 2 then second term a2 is equal to 6 and it can be written as a2 is equal to 2 into 3 which implies a2 is equal to now we know that first term a1 is equal to 2 so where we can write 2 as a1 into 3 and we also know that common ratio r is 3 so a2 is equal to a1 into similarly third term a3 which is equal to 18 can be written as 2 into 9 which can be written as 2 into 3 square as 3 square is equal to 9 which is equal to now we know that first term a1 is equal to 2 so we will write a1 into 2 ratio is so here we will write r square so a3 can be written as a1 into r square so on thus continuing like this we get enough term as an is equal to a1 into r is equal to n minus 1 where n is greater than or equal to 1 r is common ratio a1 is the sequence thus this is the required explicit formula for example suppose we are given first term a1 is equal to 2 and common ratio r is equal to minus 3 now we can find the explicit formula for this sequence using a1 is equal to a1 into r is to power n minus 1 where n is greater than or equal to 1 now we will put the values in this formula and we have the n is equal to a1 that is 2 into r that is minus 3 whole raise to power n minus 1 where n is greater than or equal to 1 now using this formula we can find any term of the sequence now let us find fifth term of the sequence so here we will put n is equal to 5 in this formula and we have a5 is equal to 2 into minus 3 whole raise to power 5 minus 1 which implies a5 is equal to 2 into minus 3 whole raise to power 4 this implies a5 is equal to 2 into 81 which implies a5 is equal to 162 so half the geometric sequence is 162 thus in general any geometric sequence is given by a1 a1 r a1 r2 a1 r2 and so on when a1 r raise to power n minus 1 and so on there this is the first term this is the second term of the geometric sequence and so on now we will find the recursive formula for the geometric sequence now here for this geometric sequence we will find the recursive formula for recursive formula we will establish the relation between the given term and its preceding or previous term now here you can see first term of the sequence that is a1 is equal to 2 then second term a2 is equal to 6 which can be written as 3 into 2 or we can write a2 as but that is the common ratio which is equal to 3 into that is the first term which is equal to 2 thus second term is product of common ratio and its preceding term that is a1 the term of the sequence 3 which is equal to a1 can be written as 3 into 6 or we can write a3 as now we know that r is equal to 3 into now here you can see second term of the sequence that is a2 is equal to 6 so here we can write a3 is equal to r into a2 now fourth term of the sequence that is a4 is equal to 54 which can be written as 3 into 18 we can write a4 now 3 is the common ratio that is r into 18 is the third term of the sequence so we will write a3 here so we have a4 is equal to r into a3 now like this t that each term is product of common ratio and its preceding term a n is equal to r into a n minus 1 where n is greater the recursive formula for the given geometric sequence is a1 is equal to 2 a n is equal to 3 into a n minus 1 where n is greater than 1 and here 3 is the common ratio in general the recursive formula for a geometric sequence is given by n is equal to r into a n minus 1 where n is greater than 1 that is a1 is known thus we can find the geometric sequence using recursive formula when we know the common ratio of sequence that is we can also derive explicit formula from the recursive formula for example we are given a n is equal to 5 into a n minus 1 a1 is equal to 2 explicit formula in geometric sequence the recursive formula is given by a n is equal to r into a n minus 1 where n is greater than 1 on comparing a n is equal to 5 into a n minus 1 with this formula we get r is equal to 5 is equal to 5 also here we are given a1 is equal to 2 that is we are given of the sequence for the geometric sequence is given by 1 into r raised to power n minus 1 where n is greater than equal to 1 sorry a little bit r is equal to 5 and a1 is equal to 2 in this formula and we get n is equal to n is greater than equal to 1 formula for the sequence so we have derived explicit formula from the given recursive formula so in this session we have discussed all to obtain recursive formula and explicit formula for geometric sequence and this completes our session hope you all have enjoyed the session