 Hello and welcome to this session. In this session we will discuss a question which says that Suppose you receive $500,000 in a lottery, this amount is being paid over 20 years time. You are given $35,000 per year from this amount. You deposit $500,000 in the bank account paying 4% annual compound interest every year and you withdraw $35,000 at the end of every year. Write a recursive formula that marries the situation. Now let us start with the solution of the given question in recursive. We find each term in the sequence by using the previous terms. When the first term or initial term or value is known, point of question we can see. The initial amount in bank is $5,000. That is, we have A0 is equal to $500,000. Now that gives 4% annual interest on it and you withdraw $35,000. Now let us find amount after one year. Let us denote it by A1. After one year the compound interest is 4%. So after one year amount after the interest will be $500,000 plus 4% of $500,000. Also you withdraw $35,000 per year from the amount. So after one year is equal to $500,000 plus 4% of $500,000 the whole minus $35,000. Now let the amount in the bank after one year be A1. So A1 is equal to now from these two terms taking $500,000 common it will be $500,000 into 1 plus 4 by 100 the whole minus $35,000. This implies A1 is equal to $100,000 into 1 plus 0.04 the whole minus $35,000 which implies A1 is equal to $500,000 into 1.04 the whole minus $35,000. Now this implies A1 is equal to now $500,000 into 1.04 is $520,000 minus this implies A1 is equal to and $85,000 in the bank after one year is equal to $85,000. Next year will be will be equal to $485,000 $85,000 minus $35,000 and $85,000 common from both these terms it will be $485,000 into 1 plus 4 by 100 the whole minus $35,000 which is equal to $485,000 into 1 plus 0.04. This implies 0.04 the whole minus $35,000 and this is equal to $485,000 into 1.04 the whole minus $35,000 and this is equal to and $400 minus $35,000 subtracting. This implies A2 is equal to $2,000 and $400 that is A2 is equal to $400. Now $69,400 amount next year will be A3 which is equal to $9,400 plus 4% of $469,400 the whole minus $35,000 which is equal to $469,400 into 1.04 the whole minus $35,000 which is equal to $469,400 into 1 plus 0,4 the whole minus $30,000. $55,488,176 minus $35,000 is equal to $53,176. So amount in the bank after three years that is $8,53,176 and the following amounts year after year first of all we have initial amount that is A0 is equal to $500,000 then we have amount after one year that is A1 is equal to $500,000 into $1.04 minus $35,000 that is equal to $85,000 is equal to now here we can see that initial amount A0 is equal to $500,000 into $1.04 minus A2 is equal to $85,000 into $1.04 minus $35,000 which is equal to now we can write it as A2 is equal to now here you can see is equal to $485,000 $1.04 minus the third year amount that is A3 which is equal to $469,400 into $1.04 minus $35,000 A3 is equal to now here you can see that $469,400 so here we can write it as A2 into $1.04 like this we see the pattern that next year's amount multiplied by the interest factor minus $35,000 you can see that the next year's amount that is A1 is equal to previous year's amount that is A0 that is $1.04 minus and see this in other cases also the amount will be is equal to A n minus 1 into $1.04 minus $35,000 that is the amount after which is A n is equal to previous year's amount that is A n minus 1 into that is $1.04 minus $35,000 the following recursive formula that is A0 is equal to $500,000 A n is equal to A n minus 1 into $1.04 minus is greater than equal to 1 so we have written the recursive formula that models the given situation and this is the solution of the given question that's all for this session hope you all have enjoyed this session