L7Sequences - Introduction to Microeconomics Lecture 7...

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1 Introduction to Microeconomics Lecture 7 Maths: Sequences, Series and Limits; the Economics of Savings & Investment Ian Crawford Outline z Sequences and series z Arithmetic and geometric z Deposits and interest z Present Value Present Value z Savings and borrowing decisions z Limits z The number e and some applications Sequences: order set of numbers z 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,… z The dots indicate that these are infinite sequences. Sequences may be finite. z 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 z A formula may use the preceding terms: “a recurrence relation”. z An example is u 1 = 1, u 2 = 1, u n = u n- 1 + u n- 2 z This gives the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13,… Series: sum of a sequence z A SERIES is formed when the terms of a sequence are added together. z Notation: 12 ... n rn uuu u +++ Notation: z Here the notation Σ (pronounced “sigma”) denotes “the sum of” 1 r = Series: sum of a sequence z Some times it’s handy to express a product as a sum. z Notation: ... ≡××× n u z Here the notation denotes “the product of” z By taking logs it becomes a sum: 1 = r 1 1 ln ln = = ⎛⎞ ⎜⎟ ⎝⎠ n n rr r r uu Arithmetic and Geometric Sequences z An ARITHMETIC SEQUENCE is one in which the difference between successive terms is constant, such as 2, 4, 6, 8,… z Here the common difference is 2 z A GEOMETRIC SEQUENCE is one in which the ratio of successive terms is a constant such as ½, 1, 2, 4, 8. . z Here the common ratio is 2
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2 Geometric series z 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 ) z 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 Application: finance z 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. z At the end of year 1 you receive 0.06 x 500 = £30 z The interest is added to the £500, so you have £530 z At the end of year 2 you receive 0.06 x £530 = £31.80, so your total sum is £561.80 z Generally: if you invest P and interest is paid annually at interest rate i then after one year you have P (1 + i ) z After two years you have P (1 + i )(1 + i ) = P (1 + i ) 2 z After t years you have P (1 + i ) t z 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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L7Sequences - Introduction to Microeconomics Lecture 7...

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