although both rules can be used in finding missing terms‚ explicit will allow an easier time finding nonconsecutive missing terms. There are also two types of sequences that recursive and explicit rules can be applied to: arithmetic and geometric. An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This difference is the common difference which the variable d is commonly used to represent it. If a sequence is arithmetic meaning it has a common
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Core 1 Linear Graphs and Equations For any straight line‚ the gradient (M) is: dy/dx or difference in y/difference in x which is (y2-y1)/(x2-x1) Equation of a line: y=mx+c which is used when the gradient and intercept is known or y-y1=m(x-x1) when the gradient and the co-ordinates (x1‚y1) of a single point that the line passes through is known. You’ll need to learn this equation. [The equation of the line can be kept in this form unless stated in the exam. (reduces error chance) Also
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Consider the arithmetic series 2 + 5 + 8 +.... (a) Find an expression for Sn‚ the sum of the first n terms. (b) Find the value of n for which Sn = 1365. (Total 6 marks) 11. Find the sum to infinity of the geometric series (Total 3 marks) 12. The first and fourth terms of a geometric series are 18 and respectively. Find (a) the sum of the first n terms of the series; (4) (b) the sum to infinity of the series. (2) (Total 6 marks) 13. Find the coefficient of x7 in the expansion
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solution as the two examples displayed. For instance‚ 3x + 4y = 5 and x -2y = -5‚ another system‚ also displays the same pattern as the first set and has a solution of (-1‚ 2). Essentially‚ this pattern is indicating an arithmetic progression sequence. Arithmetic progression is described as common difference between sequences of numbers. In a specific sequence‚ each number accordingly is labelled as an. the subscript n is referring to the term number‚ for instance the 3rd term is known as a3. The formula
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below shows some of the formulae entered to generate the spreadsheet above. Extrapolation in terms of a diagram and geometric progressions T8 T16 “T32”“ T64” X According to the theory derived earlier 32 16 16 8 1 ( - 4 T T≈ + T T ) This gives us the so called “extrapolated” value 32 16 16 8 1 " " ( -). 4 T T TT = + Note‚ this is exactly how “T32” was calculated on the previous page. And then 2 2 64 32 16 8 16 16 8 16 8 1 11
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Topics Topic 1—Algebra 1.1 The nth term of an arithmetic sequence The sum of n terms of an arithmetic sequence The nth term of a geometric sequence un = u1 + (n − 1)d S n= n n (2u1 + (n − 1)d ) = (u1 + un ) 2 2 un = u1r n −1 The sum of n terms of a u1 (r n − 1) u (1 − r n ) ‚ r ≠1 = = 1 Sn finite geometric sequence r −1 1− r The sum of an infinite geometric sequence 1.2 Exponents and logarithms Laws of logarithms S∞ = u1 ‚ r 0 − cos ∫ sin x dx = x + C = ∫ cos x dx sin x + C ∫e
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Thomas Malthus Principles of Population Today‚ there is both agreement and disagreement of Thomas Malthus’ essay on the principles of population. Malthus stated that population grows exponentially or at “geometric rate” and food production grows at arithmetic rate‚ or linearly. Geometric rate grows in a series of numbers (2‚4‚8‚16‚32…etc.)‚ which shows that children will grow up and each have their own children‚ and those children will have their own children. Eventually the base numbers of children
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Rock Harmony (Andy Goldsworthy: “Rock Creation”) Richard Lipoczi Art 100 Jennifer Monroe 04/25/2013 Andy Goldsworthy is a British naturalist artist‚ mostly sculptor and photographer‚ creating in the twentieth and twenty-first centuries. He is still alive in these days. All his works draw on their themes from the nature: he uses only natural materials in their original environments to his works. He expresses his thoughts through transforming the natural matters to certain kinds of
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A population of 80 longhorn cattle was introduced onto an island on 1 October 1960. There was no subsequent migration of longhorn cattle to or from the island. Let Pn denote the size of the longhorn cattle population on the island on 1 October n years after 1960. (a) In this part assume that‚ for each integer n ≥ 0‚ the number of births and deaths in the year beginning 1 October n years after 1960 are 1.52Pn and 1.24Pn‚ respectively. (i) Find a recurrence system satisfied by Pn. (ii) State
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Geometric and Arithmetic Sequences to Questions 35 & 37 MAT-126: Survey of Mathematical Methods(ACO1141A) October 11‚ 2011 As one observes an arithmetic sequence‚ it is imperative to use inductive and deductive reasoning to use the right mathematical approach of geometric or arithmetic sequence to solve the equation in the most pragmatic way. Most times both inductive and deductive reasoning is used on an equation or variable to come up with the most direct approach to an answer
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