 Hello and welcome to the session. In this session, we will discuss how to write an arithmetic sequence both recursively and with an explicit formula and translate between the two forms. Now, let us find the recursive formula for an arithmetic sequence. Now, consider the following sequence. Here, first term that is a1 is equal to first, second term a2 is equal to 7, third term a3 is equal to 10 and so on. Now, let us see the difference between the two consecutive terms. Now, a2 minus a1 is equal to 7 minus 4 that is equal to 3. a3 minus a2 is equal to 10 minus 7 that is equal to 3. Now, here a4 is 13. So, a4 minus a3 is equal to 13 minus 10 that is also 3 and so on. Thus, there is a common difference denoted by d between the consecutive terms of the sequence and that difference is d is equal to 3. So, we have the following pattern a1 is equal to 4 then a2 is equal to 7 which can be written as 4 plus 3 or we can write it as a2 is equal to now a1 is equal to 4 so a2 can be written as a1 plus 3 then a3 is equal to 10 which can be written as 7 plus 3 or we can write it as a3 is equal to now a2 is equal to 7 so a3 can be written as a2 plus 3 like this is sum of the preceding term and the common difference 3 thus nth term of the sequence will be an is equal to an minus 1 plus 3 where an minus 1 is preceding or previous term of an recursive formula for this sequence is given by a1 is equal to 4 an is equal to an minus 1 plus 3 where n is greater than 1 write it as a1 is equal to 4 an plus 1 is equal to an plus 3 where n is greater than equal to 1 thus in general the recursive formula for alphabetic sequence is given by an plus 1 is equal to an plus d where n is greater than equal to 1 and d is common difference write it as an is equal to an minus 1 plus d where n is greater than 1 also that is a1 is known now let us see an example consider the following sequence now here common difference d is equal to 2 now here you can see difference between 2 consecutive terms of this sequence that is 4 minus 2 is 2, 6 minus 4 is 2, 8 minus 6 is 2 and so on so here common difference d is equal to 2 is equal to where n is greater than that is a1 is equal to 2 the value of d in this formula and we get an plus 1 is equal to an is equal to 2 where n is greater than equal to 2 now let us find explicit formula for this alphabetic sequence now here common difference d is equal to 3 now here you can see 7 minus 4 is 3, 10 minus 7 is 3, 30 minus 10 is again 3 that is a1 is equal to 4 which can be written as 4 plus 0 into 3 term a2 is equal to 7 which can be written as 4 plus 1 into 3 which can also be written as simply a1 plus d then that is the third term is 10 which can be written as 4 into 3 and this can be written as into d or simply a1 plus 2d further now this a4 is equal to 13 which can be written as 4 plus 3 into 3 and this can be written as a1 plus 3 into d or simply a1 plus 3d term of this sequence an is equal to a1 into d so here formula is equal to now a1 here is 4 whole into d which is n is equal to 4 plus an is equal to now this 1 plus 3n or we can write this an is equal to 3n plus 1 n is equal to 3n plus 1 where n is greater than equal to 1 thus in an alphabetic sequence we are finding all the terms of the sequence when we know the first term a1 and the common difference d is given by equal to a1 plus n minus 1 the whole into d where n is greater than equal to 1 now suppose we have to find the 11th term of the sequence whose explicit formula is given as an is equal to 3n plus 1 where n is greater than equal to 1 so here 11 put n is equal to 11 and this formula and we have a11 is equal to 3 into 11 plus 1 which implies a11 is equal to 33 plus 1 which implies a11 is equal to 34 the sequence is 34 the formula from the requested formula so formula is equal to 4 an is equal to an minus 1 plus 3 where now see here d is equal to 3 and is equal to 4 1 the whole into d which will be equal to 1 the whole into d is 3 which is equal to 4 plus 3n minus 3 which is equal to 3n plus 1 equal to 1 is a formula from the requested formula to obtain the reference formula an alphabetic sequence and this completes our session hope you all have enjoyed the session