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Fibonacci Sequence

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  Fibonacci Sequence The Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... The next number is found by adding up the two numbers before it: the 2 is found by adding the two numbers before it (1+1), the 3 is found by adding the two numbers before it (1+2), the 5 is (2+3), and so on! Example: the next number in the sequence above is 21+34 =   55 It is that simple! Here is a longer list: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, ... Makes A Spiral When we make squares with those widths, we get a nice spiral: Do you see how the squares fit neatly together? For example 5 and 8 make 13, 8 and 13 make 21, and so on. This spiral is found in nature! See:  Nature, The Golden Ratio, and Fibonacci The Rule The Fibonacci Sequence can be written as a "Rule" (see  Sequences and Series ). First, the terms are numbered from 0 onwards like this: n ...
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CONCENTRIC AND CONGRUENT CIRCLES

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  CONCENTRIC AND CONGRUENT CIRCLES CIRCLE: A circle is the set of points in a plane that are equidistant from a given point. CONCENTRIC CIRCLE: Two or more circles that have the same center,but different radii are known as concentric circles. Concentric Circle Examples In Real Life Have you ever observed the wheel of a ship? It has two concentric circles joined by the spokes. Many of us have played darts. A dartboard also has concentric circles. The circles in the dartboard are drawn around a common center, that is, the bull's eye. CONGRUENT CIRCLE: Two or more circles with the same radius,but different centres are known as congruent circles.
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ALGEBRA

  Algebra is a branch of mathematics that substitutes letters for numbers. Important Formulas in Algebra Here is a list of Algebraic formulas  – a 2  – b 2  = (a – b)(a + b) (a + b) 2  = a 2  + 2ab + b 2 a 2  + b 2  = (a + b) 2  – 2ab (a – b) 2  = a 2  – 2ab + b 2 (a + b + c) 2  = a 2  + b 2  + c 2  + 2ab + 2bc + 2ca (a – b – c) 2  = a 2  + b 2  + c 2  – 2ab + 2bc – 2ca (a + b) 3  = a 3  + 3a 2 b + 3ab 2  + b 3  ; (a + b) 3  = a 3  + b 3  + 3ab(a + b) (a – b) 3  = a 3  – 3a 2 b + 3ab 2  – b 3  = a 3  – b 3  – 3ab(a – b) a 3  – b 3  = (a – b)(a 2  + ab + b 2 ) a 3  + b 3  = (a + b)(a 2  – ab + b 2 )
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