 Hi and welcome to the session. Let us solve the following question that says, find the sum of the following series up to n terms 0.6 plus 0.66 plus 0.66 plus so on. So let's start with the solution and let us denote the sum of n terms by Sn, which is equal to 0.6 plus 0.66 plus 0.66 plus so on to n terms. Now let us take 6 common. So inside the packet we have 0.1 plus 0.11 plus 0.111 plus so on to n terms. Now this is further equal to 6 into 1 upon 10 plus 11 upon 100 plus 111 upon 1000 plus to n terms. Now multiplying the numerator and denominator by 9, we have 6 upon 9 into 9 upon 10 plus 99 upon 100 plus 999 upon 1000 plus so on to n terms which is further equal to 6 upon 9 and 9 upon 10 can be written as 1 minus 1 upon 10 plus 99 upon 100 can be written as 1 minus 1 upon 100 plus 99 upon 1000 can be written as 1 minus 1 upon 1000 and so on to n terms. So that implies that 6 upon 9 into 1 plus 1 plus 1 n times minus 6 upon 9 into 1 upon 10 plus 1 upon 10 square plus 1 upon 10 cube plus so on to n terms further equal to 6 upon 9 into n minus 6 upon 9. Now this is a GP series whose ratio is 1 upon 10 which is less than 1 and the first term is A and if we have a GP series A, AR, AR square and so on, the sum of these n terms is equal to A into 1 minus r raised to the power n upon 1 minus r if r is less than 1. A is the first term and r is the common ratio. Now here this is a GP where A is the first term, so A is 1 upon 10 and the common ratio is again 1 upon 10 so this can be written as 1 upon 10 into 1 minus 1 upon 10 raised to the power n upon 1 minus 1 upon 10. So this will be equal to 6 upon 9 n minus 6 upon 9 into 1 upon 10 into 10 upon 9 into 1 minus 1 upon 10 raised to the power n. Now 10 cancels out with 10 and here we have 2 upon 3, here again 2 upon 3 so we have 2 upon 3 into n minus 2 upon 3 into 1 upon 9 and inside the bracket 1 minus 10 raised to the power minus n which is further equal to 2 upon 3 into n minus 2 upon 27 into 1 minus 10 raised to the power minus n, thus the sum of the given series is 2n upon 3 minus 2 upon 27 into 1 minus 10 raised to the power minus n. So this completes the session, take care and have a good day.