 Hello and welcome to the session. In this session we will learn about amount or future value of an NUD. Now the amount or future value of an NUD is the sum of the compound amounts of all the payments accumulated at the end of the term. Now let capital A denotes the future amount of an NUD and let A dollars be the annual payment of each installment of an NUD of N periods and R percent be the rate of interest per period and R denotes the interest. One dollar for the same period therefore R is equal to R upon 100. Now let us discuss the amount or future value of an NUD in case of NUD due. Now in case of an NUD due the payments are made at the beginning of each payment period. Now suppose two payments made for N periods the first payment the second payment is interest for minus one the whole period, interest for one period. Now we can find out the amount by using this formula which we are using in case of the compound interest. Now here as I is equal to R upon 100 so the amount can be calculated by using the formula P into 1 plus I the whole raised to power N where P is the principle for each period. Now here in case of NUD due the first payment earns an interest for N periods that is in this case the payments are made at the beginning of each payment period. So this payment that is at the beginning of the first period will earn an interest for N periods so the first payment amounts to N into 1 plus I whole raised to power N. Now the second payment that is at the end of the first period and at the beginning of the second period will earn interest for N minus one periods. So the second payment amount to N into 1 plus I whole raised to power N minus one. Now the third payment will earn interest for periods so the third payment amounts to 1 plus I whole raised to power N minus two and continuing likewise will be made to periods that is at the beginning of the Nth period and the Nth payment amounts to into 1 plus I the whole. That is interest for one periods only. Now the sum adds the amount of the NUD that is the future amount of the NUD which is denoted by capital A. Therefore the amount NUD that is capital A is equal to A into 1 plus I whole raised to power N plus A into 1 plus I whole raised to power N minus one plus so on up to plus A into 1 plus I the whole. Now this is equal to taking A into 1 plus I the whole common from all these terms it will be A into 1 plus I the whole into 1 plus I this whole raised to power N minus one plus 1 plus I this whole raised to power N minus two plus so on up to plus 1 plus I the whole plus one and this complete whole. Now in the geometric progression the sum of the N terms that is fN is equal to A into R raised to power N minus one the whole will upon R minus one where R is written in one. Now this is also a geometric progression where 1 plus I the whole and A is equal to one. So by using this formula this will be equal to A into 1 plus I the whole into 1 plus I N minus one whole upon 1 plus I the whole minus one this complete whole which is equal to 1 plus I the whole minus one this complete whole the whole future amount of the energy which is denoted by capital N denotes the last payment made after N minus one periods and the accumulation money is received after N periods in the case of record deposits in a bank. The future amount of the energy in case of immediate energy. Now in this case the payments are made at the end of each payment period therefore at the end of first interest for N minus one the whole the whole periods that is A will on an interest for N minus one periods into 1 plus I the whole raised to power N minus one. Now the second installment will earn an interest for N minus two periods into one plus I whole raised to power N minus two. The third installment for N minus three the last installment the interest since its payment is made at the end of the term so it will remain the compound amount since will give us the future amount of the energy. Therefore the amount the energy which we have denoted as capital A is equal to A into one plus I the whole raised to power N minus one plus A into one plus I whole raised to power N minus two plus so on up to plus A into one plus I the whole plus A into one plus I whole raised to power zero. Now here as one plus I whole raised to power zero will be one A common two equal to A into one plus I whole raised to power N minus one plus one plus I whole raised to power N minus two up to plus one plus I the whole. Now again using the formula but finding out the sum up to N terms that is S N in a geometric progression where R is given as this will be equal to A into one into one plus I the whole raised to power N minus one and this computer and the whole minus one. And this complete whole is one plus I the whole. Now this is equal to into one plus I the whole raised to power N minus one and this complete whole. Now in case of immediate energy this is the future amount of the energy which is denoted by capital A. And here in this case N denotes the last element made after N periods as in the case of one taken from the bank. So in this session we have learnt about N F and N U T. And this completes our session. Hope you all have enjoyed the session.