# (Solved) : Suppose A0 Dollars Deposited Bank Pays 5 Interest Per Year Compounded Quarterly One Quarte Q41474475 . . .

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Suppose a0 dollars are deposited in a bank that pays 5% interestper year, compounded quarterly. After one quarter the value of theaccount is **a0 × (1 + (0.05)/4)** dollars. At the endof the second quarter, the bank pays interest not only on theoriginal amount a0 but also on the interest earned in the firstquarter; thus the value of the investment at the end of the secondquarter is **[a0 × (1 + (0.05)/4)] × (1 + (0.05)/4) = a0 × (1+ (0.05)/4)2** dollars. At the end of the thirdquarter, the bank pays interest on this amount so that the accountis now worth **a0 × (1 + (0.05)/4)3**dollars, and at the end of the whole year the investment is finallyworth **a0 × (1 + (0.05)/4)4** dollars. Ingeneral, if a0 dollars are deposited at an annual interest rate x,compounded n times per year, then the account value after one yearis **a0 × In(x) where In(x) = (1 +x/n)n**.

This is the compound interest formula. It is well known that forfixed x, **lim n→∞ In(x)=ex**.

(a) Determine the relative condition number for the problem ofevaluating. For x = 0.05, would you say that this problem iswell-conditioned or ill-conditioned?

(b) Use MATLAB to compute for x = 0.05 and for n = 1, 10,102, . . . , 1015. Use ‘format long e’ sothat you can see if your results are converging to ex,as one might expect. Turn in a table with your results and alisting of the MATLAB command(s) you used to compute theseresults.

(c) Try to explain the results of part (b). In particular, for n =1015, you probably computed 1 as your answer. Explainwhy. To see what is happening for other values of n, consider theproblem of computing zn, where z = (1 + x/n) when n islarge. What is the relative condition number of this problem? Ifyou make an error of about 10-16 in computing z, aboutwhat size error would you expect in zn?

(d) Can you think of a better way than the method you used in part(b) to accurately compute for x = 0.05 and for large values of n?Demonstrate your new method in MATLAB or explain why it should givemore accurate results.

## Expert Answer

Answer to Suppose a0 dollars are deposited in a bank that pays 5% interest per year, compounded quarterly. After one quarter the v…