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Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

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BuyFindarrow_forward

Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

Retirement Accounts Many college professors keep retirement savings with TIAA, the largest annuity program in the world. Interest on these accounts is compounded and credited daily. Professor Brown has $ 275 , 000 on deposit with TIAA at the start of 2015 and receives 3.65 % interest per year on his account.

(a) Find a recursive sequence that models the amount A n in his account at the end of the n th day of 2015 .

(b) Find the first eight terms of the sequence A n , rounded to the nearest cent.

(c) Find a formula for A n .

To determine

(a)

To find:

A recursive sequence that models the amount An in account at the end of the nth day of 2015.

Answer

Solution:

The recursive sequence that models the amount An in account at the end of the nth day of 2015 is An=1.0001×An1

Explanation

Given:

The original deposit amount is $275000 and interest rate is 3.65% per year.

Interest is compounded and credited daily.

Approach:

Since the interest is compounded and credited daily, the interest rate for each day is:

(3.65365)%=0.01%

The formula hence used will be:

(Amount at the end of day)=(Amount at the end of the previous day)+0.0001×(Amount at the end of the previous day)

Generalize the above formula for n days to obtain the following formula:

(Amount at the end of nth day)=1.0001×(Amount at the end of the previous day)

Calculation:

As explained above,

(Amount at the end of nth day)=1.0001×(Amount at the end of the previous day)

If An is the amount at the end of the nth day, then the recursive formula becomes:

An=(1.0001)×An1, A0=$275000

Therefore, a recursive sequence that models the amount An in account at the end of the nth day of 2015 is An=(1.0001)×An1, A0=$275000.

Conclusion:

Hence, a recursive sequence that models the amount An in account at the end of the nth day of 2015.is An=1.0001×An1, A0=$275000.

To determine

(b)

To find:

The first eight terms of the sequence An.

Answer

Solution:

The first eight terms of the sequence An are:

A0=275000.00A1=275027.50A2=275055.00A3=275082.51

A4=275110.02A5=275137.53A6=275165.04A7=275192.56

Explanation

Given:

The initial amount is 275000.

Approach:

Use recursive relation calculated in part (a).

Calculation:

From part (a), the recursive formula is:

An=(1.0001)×An1, A0=$275000 ……(1)

Substitute n=1, 2, 3, 4, 5, 6, 7 and 8 successively in formula (1),

Therefore, the first eight terms of the sequence An are:

A0=275000.00A1=1.0001×275000=275027.50

A2=1.0001×275027.50=275055.00A3=1.0001×275055.00=275082.51

A4=1.0001×275082.51=275110.02A5=1.0001×275110.02=275137.53

A6=1.0001×275137.53=275165.04A7=1.0001×275165.04=275192.56

Conclusion:

Hence, the first eight terms of the sequence An are:

A0=275000.00A1=275027.50A2=275055.00A3=275082.51

A4=275110.02A5=275137.53A6=275165.04A7=275192.56

To determine

c)

To find:

A formula for An.

Answer

Solution:

A formula for An is An=275000(1.0001)n.

Explanation

Given:

The initial amount is 275000.

Approach:

Use part (b)

Calculation:

From the pattern,

A0=275000.00=(1.0001)0×275000A1=275027.50=(1.0001)1×275000

A2=(1.0001)2×275000A3=(1.0001)3×275000A4=(1.0001)4×275000A5=(1.0001)5×275000

A6=(1.0001)6×275000A7=(1.0001)7×275000

Therefore, a formula for An is An=275000(1.0001)n.

Conclusion:

Hence, a formula for An is An=(1.0001)n×An1.

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