SECTION 12.1 Sequences 863 When deposits are made at the same time that the interest is credited, the annuity is called ordinary . We will only deal with ordinary annuities here. The amount of an annuity is the sum of all deposits made plus all interest paid. Suppose that the initial amount deposited in an annuity is M$ , the periodic deposit is P$ , and the per annum rate of interest is r% (expressed as a decimal) compounded N times per year.The periodic deposit is made at the same time that the interest is credited, so N deposits are made per year. The amount An of the annuity after n deposits will equal − A , n 1 the amount of the annuity after −n 1 deposits, plus the interest earned on this amount, plus P, the periodic deposit. That is, ( ) = + + = + + − − − A A r N A P r N A P 1 n n n n 1 1 1 ↑ Amount after n deposits ↑ Amount in previous period ↑ Interest earned ↑ Periodic deposit We have established the following result: THEOREM Annuity Formula If = A M 0 represents the initial amount deposited in an annuity that earns r% per annum compounded N times per year, and if P is the periodic deposit made at each payment period, then the amount An of the annuity after n deposits is given by the recursive sequence ( ) = = + + ≥ − A M A r N A P n 1 1 n n 0 1 (9) Formula (9) may be explained as follows: the money in the account initially, A ,0 is M$ ; the money in the account after −n 1 payments, − A , n 1 earns interest − r N An 1 during the n th period; so when the periodic payment of P dollars is added, the amount after n payments, A ,n is obtained. Saving for Spring Break A trip to Cancun during spring break will cost $1250 and full payment is due March 2. To have the money, a student, on September 1, deposits $500 in a savings account that pays 2% per annum compounded monthly. On the first of each month, the student deposits $100 in this account. (a) Find a recursive sequence that explains how much is in the account after n months. (b) Use the TABLE feature to list the amounts of the annuity for the first 6 months. (c) After the deposit on March 1 is made, is there enough in the account to pay for the Cancun trip? (d) If the student deposits $125 each month, will there be enough for the trip after the March 1 deposit? EXAMPLE 10 Solution (a) The initial amount deposited in the account is = A $500. 0 The monthly deposit is = P $100, and the per annum rate of interest is = r 0.02 compounded = N 12 times per year. The amount An in the account after n monthly deposits is given by the recursive sequence ( ) ( ) = = + + = + + − − A A r N A P A 500 1 1 0.02 12 100 n n n 0 1 1 (b) In SEQuence mode on a TI-84 Plus CE, enter the sequence { } An and create Table 4. On September 1 ( ) = n 0 , there is $500 in the account. After the first payment on October 1, the value of the account is $600.83. After the second payment on November 1, the value of the account is $701.83. After the third payment on December 1, the value of the account is $803, and so on. Table 4
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