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L7Sequences

# L7Sequences - Introduction to Microeconomics Lecture 7...

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Introduction to Microeconomics Lecture 7 Maths: Sequences, Series and Limits; the Economics of Savings & Investment Ian Crawford

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Outline Sequences and series Arithmetic and geometric Deposits and interest Present Value Savings and borrowing decisions Limits The number e and some applications
Sequences: order set of numbers A SEQUENCE is a set of numbers arranged in a definite order, e.g. (i) 2, 4, 6, 8,…; (ii) 1, 4, 9, 16, 25,… The dots indicate that these are infinite sequences. Sequences may be finite. There may be a formula that defines the terms e.g. for sequence (ii) the n th term, u n , is u n = n 2 A formula may use the preceding terms: “a recurrence relation”. An example is u 1 = 1, u 2 = 1, u n = u n- 1 + u n- 2 This gives the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13,…

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Series: sum of a sequence A SERIES is formed when the terms of a sequence are added together. Notation: Here the notation Σ (pronounced “sigma”) denotes “the sum of” 1 2 1 ... n r n r u u u u = + + +
Series: sum of a sequence Some times it’s handy to express a product as a sum. Notation: Here the notation ∏ denotes “the product of” By taking logs it becomes a sum: 1 2 1 ... = × × × n r n r u u u u 1 1 ln ln = = n n r r r r u u

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Arithmetic and Geometric Sequences An ARITHMETIC SEQUENCE is one in which the difference between successive terms is constant, such as 2, 4, 6, 8,… Here the common difference is 2 A GEOMETRIC SEQUENCE is one in which the ratio of successive terms is a constant such as ½, 1, 2, 4, 8.. Here the common ratio is 2
Geometric series What is the sum of the first n terms in a geometric sequence? Assume the common ratio r 1 S n = a + ar + ar 2 + ar 3 + … + ar n- 1 = a (1 – r n )/(1 – r ) Proof (Workbook, Chapter 3, Sect. 2.5) S n = a + ar + ar 2 + ar 3 + … + ar n- 1 rS n = ar + ar 2 + ar 3 + … + ar n- 1 + ar n S n rS n = a – ar n which then gives the formula

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Application: finance You have £500 in the bank and the interest rate is 6% (i.e. 0.06) per year. Interest is paid at the end of the year. At the end of year 1 you receive 0.06 x 500 = £30 The interest is added to the £500, so you have £530 At the end of year 2 you receive 0.06 x £530 = £31.80, so your total sum is £561.80 Generally: if you invest P and interest is paid annually at interest rate i then after one year you have P (1 + i ) After two years you have P (1 + i )(1 + i ) = P (1 + i ) 2 After t years you have P (1 + i ) t The amount of money in your bank at the end of each year is a geometric sequence with common ratio (1 + i )
How much will pension contributions be worth?

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