Gregory Would Like To Purchase His Own Aircraft. He Borrows Some Money From A Bank To Purchase A Second-hand Plane. The Recurrence Relation Below Models The Value Of His Loan After $n$ Months, $V_n$:$[ V_0 = 58000, \quad

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Understanding the Recurrence Relation for Gregory's Loan

Gregory is an avid pilot who has always dreamed of owning his own aircraft. To make this dream a reality, he has borrowed a significant amount of money from a bank to purchase a second-hand plane. The loan amount is substantial, and Gregory needs to understand how the value of his loan will change over time. In this article, we will explore a recurrence relation that models the value of Gregory's loan after n months, Vn.

The recurrence relation for the value of Gregory's loan after n months is given by:

V0 = 58000, Vn = 0.9Vn-1

This relation states that the value of the loan after n months is 90% of the value of the loan after (n-1) months. In other words, the loan amount decreases by 10% each month.

Understanding the Recurrence Relation

To understand the recurrence relation, let's break it down step by step.

  • The initial value of the loan, V0, is 58000.
  • The recurrence relation states that the value of the loan after n months, Vn, is 90% of the value of the loan after (n-1) months, Vn-1.
  • This means that each month, the loan amount decreases by 10% of its previous value.

Solving the Recurrence Relation

To solve the recurrence relation, we can use an iterative approach. We will start with the initial value of the loan, V0, and then apply the recurrence relation repeatedly to find the value of the loan after each month.

V1 = 0.9V0 = 0.9(58000) = 52200 V2 = 0.9V1 = 0.9(52200) = 46980 V3 = 0.9V2 = 0.9(46980) = 42228 V4 = 0.9V3 = 0.9(42228) = 37945.2 V5 = 0.9V4 = 0.9(37945.2) = 34160.08 V6 = 0.9V5 = 0.9(34160.08) = 30736.07 V7 = 0.9V6 = 0.9(30736.07) = 27684.46 V8 = 0.9V7 = 0.9(27684.46) = 24915.61 V9 = 0.9V8 = 0.9(24915.61) = 22433.74 V10 = 0.9V9 = 0.9(22433.74) = 20190.67 V11 = 0.9V10 = 0.9(20190.67) = 18137.6 V12 = 0.9V11 = 0.9(18137.6) = 16313.44 V13 = 0.9V12 = 0.9(16313.44) = 14651.19 V14 = 0.9V13 = 0.9(14651.19) = 13166.31 V15 = 0.9V14 = 0.9(13166.31) = 11849.48 V16 = 0.9V15 = 0.9(11849.48) = 10645.03 V17 = 0.9V16 = 0.9(10645.03) = 9550.05 V18 = 0.9V17 = 0.9(9550.05) = 8615.04 V19 = 0.9V18 = 0.9(8615.04) = 7743.46 V20 = 0.9V19 = 0.9(7743.46) = 6971.2 V21 = 0.9V20 = 0.9(6971.2) = 6274.08 V22 = 0.9V21 = 0.9(6274.08) = 5646.37 V23 = 0.9V22 = 0.9(5646.37) = 5080.13 V24 = 0.9V23 = 0.9(5080.13) = 4572.12 V25 = 0.9V24 = 0.9(4572.12) = 4118.3 V26 = 0.9V25 = 0.9(4118.3) = 3697.07 V27 = 0.9V26 = 0.9(3697.07) = 3328.63 V28 = 0.9V27 = 0.9(3328.63) = 2994.06 V29 = 0.9V28 = 0.9(2994.06) = 2696.45 V30 = 0.9V29 = 0.9(2696.45) = 2424.4 V31 = 0.9V30 = 0.9(2424.4) = 2178.36 V32 = 0.9V31 = 0.9(2178.36) = 1959.22 V33 = 0.9V32 = 0.9(1959.22) = 1763.0 V34 = 0.9V33 = 0.9(1763.0) = 1587.27 V35 = 0.9V34 = 0.9(1587.27) = 1428.44 V36 = 0.9V35 = 0.9(1428.44) = 1285.19 V37 = 0.9V36 = 0.9(1285.19) = 1155.57 V38 = 0.9V37 = 0.9(1155.57) = 1040.1 V39 = 0.9V38 = 0.9(1040.1) = 936.09 V40 = 0.9V39 = 0.9(936.09) = 840.66 V41 = 0.9V40 = 0.9(840.66) = 756.59 V42 = 0.9V41 = 0.9(756.59) = 680.31 V43 = 0.9V42 = 0.9(680.31) = 612.28 V44 = 0.9V43 = 0.9(612.28) = 551.05 V45 = 0.9V44 = 0.9(551.05) = 494.49 V46 = 0.9V45 = 0.9(494.49) = 445.04 V47 = 0.9V46 = 0.9(445.04) = 400.46 V48 = 0.9V47 = 0.9(400.46) = 360.41 V49 = 0.9V48 = 0.9(360.41) = 324.37 V50 = 0.9V49 = 0.9(324.37) = 291.83 V51 = 0.9V50 = 0.9(291.83) = 262.95 V52 = 0.9V51 = 0.9(262.95) = 236.46 V53 = 0.9V52 = 0.9(236.46) = 212.61 V54 = 0.9V53 = 0.9(212.61) = 190.75 V55 = 0.9V54 = 0.9(190.75) = 171.27 V56 = 0.9V55 = 0.9(171.27) = 153.83 V57 = 0.9V56 = 0.9(153.83) = 138.75 V58 = 0.9V57 = 0.9(138.75) = 124.88 V59 = 0.9V58 = 0.9(124.88) = 112.1 V60 = 0.9V59 = 0.9(112.1) = 100.99 V61 = 0.9V60 = 0.9(100.99) = 90.89 V62 = 0.9V61 = 0.9(90.89) = 81.8 V63 = 0.9V62 = 0.9(81.8) = 73.22 V64 = 0.9V63 = 0.9(73.22) = 65.9 V65 = 0.9V64 = 0.9(65.9) = 59.01 V66 = 0.9
Frequently Asked Questions about Gregory's Loan Recurrence Relation

A: The initial value of Gregory's loan, V0, is 58000.

A: The recurrence relation models the value of Gregory's loan after n months as Vn = 0.9Vn-1, where Vn is the value of the loan after n months and Vn-1 is the value of the loan after (n-1) months.

A: The loan amount decreases by 10% each month.

A: We can solve the recurrence relation by applying it repeatedly, starting with the initial value of the loan, V0. This will give us the value of the loan after each month.

A: The value of the loan after 1 month, V1, is 52200.

A: The value of the loan after 2 months, V2, is 46980.

A: Yes, we can use the recurrence relation to find the value of the loan after any number of months. We simply need to apply the recurrence relation repeatedly, starting with the initial value of the loan, V0.

A: We can use the recurrence relation to make predictions about the future value of the loan by applying it repeatedly, starting with the current value of the loan. This will give us the value of the loan after each subsequent month.

A: The recurrence relation has many potential applications in real-world scenarios, such as:

  • Modeling the value of a loan over time
  • Predicting the future value of an investment
  • Analyzing the impact of interest rates on loan values
  • Developing strategies for managing debt

A: Yes, we can use the recurrence relation to compare the value of different loans. By applying the recurrence relation to each loan, we can see how the value of each loan changes over time.

A: We can use the recurrence relation to make informed decisions about loan management by analyzing the impact of different interest rates, loan terms, and repayment strategies on the value of the loan.

A: Some potential limitations of the recurrence relation include:

  • The assumption that the interest rate remains constant over time
  • The assumption that the loan amount decreases by 10% each month
  • The lack of consideration for other factors that may impact the value of the loan, such as inflation or market fluctuations.

A: Yes, we can use the recurrence relation to model other types of financial instruments, such as investments or savings accounts. By modifying the recurrence relation to reflect the specific characteristics of the instrument, we can use it to make predictions about its future value.