An example of such a sequence would be 5, 12, 19, 26, 33…, where and . This is an increasing arithmetic sequence, as the terms are increasing. Decreasing arithmetic sequences have . 6. 3 Geometric sequences also start with any number a (though usually a is nonzero here), but this time we are not adding an extra d value each time- we multiply a by a factor of r. Thus, the term is tn . Geometric sequences can either be monotonic, when r is positive and the terms are moving in one direction, or alternating, where and the terms alternate between positive and negative values, depending on n. . 4 Recursion is the process of choosing a starting term and repeatedly applying the same process to each term to arrive at the following term. Recursion requires that you know the value of the term immediately before the term you are trying to find. A recursive formula always has two parts: 30 MCR3U Exam Review 1. the starting value for a1. 2. the recursion equation for an as a function of an-1 (the term before it. ) Recursive formula: Same recursive formula: 6. 5 A series is a sequence of numbers that represent partial sums for another sequence.
For example, if my sequence is 1,2,3,4… then my series would be 1,1+2,1+2+3,… , or 1,3,6,10…. Arithmetic Series Formula: 6. 6 Geometric Series Formula: Binomial Theorem Pascal’s Triangle”. To make the triangle, you start with a pyramid of three 1’s, like this: Then you get the next row of numbers by adding the pairs of numbers from above. (Where there is only one number above, you just carry down the 1. ) Keep going, always adding pairs of numbers from the previous row.. the power of the binomial is the row that you want to look at. The numbers in the row will be your coefficients
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The powers on each term in the expansion always added up to whatever n was, and that the terms counted up from zero to n. Ex. Expand (x2 + 3)6 (1)(x12)(1) + (6)(x10)(3) + (15)(x8)(9) + (20)(x6)(27) + (15)(x )(81) + (6)(x )(243) + (1)(1)(729) = x12 + 18×10 + 135×8 + 540×6 + 1215×4 + 1458×2 + 729 1 e. g. 8 0 3 4 2 *Coefficients are from P’s Triangle 6 0 31 MCR3U Exam Review Prime Factorization Factor a number into its prime factors using the tree diagram method. 6 1 Evaluate. e. g. 0 e. g. Simplify. ?2 (3 0 + 32 ) ?2 = (1 + 9 ) ?2 2 = 10 1 = 2 3 10 1 2 = 100 Follow the order of operations.
Evaluate brackets first. ? b3 ? ? ? 3 ? ? 2a ? ? ? ?2 b 3 ( ? 2 ) = (2a ? 3 ) ? 2 b ? 6 = ? 2 ? 3( ? 2 ) 2 a 2 2 b ? 6 = a6 4 = 6 6 ab Power of a quotient. Power of a product. 5 Exponential Functions f(x) = 2x 6 4 In general, the exponential function is defined by the equation, y=a x f(x) = 2x-2+3 6 4 2 or , f ( x ) = a x a > 0, x ? R . 2 Compound Interest Calculating the future amount: A = P (1 + i ) Calculating the present amount: P = A(1 + i ) ? n n A – future amount P – present (initial) amount i – interest rate per conversion period n – number of conversion periods