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Aim: Summation Notation Course: Alg. 2 & Trig. Do Now: Aim: What is this symbol It’s Greek to me! Find the sum of the geometric series.

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Aim: Summation Notation Course: Alg. 2 & Trig. Definition of Summation Notation sigma sum of terms The sum of the first n terms of a sequence is represented by When n is a specific number, the sum of a sequence is called a finite series. where i is the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation.

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Aim: Summation Notation Course: Alg. 2 & Trig. Summation Notation where i is the index of summation n is the upper limit of summation 0 is the lower limit of summation The first term of the summation is formed by substituting the lower limit for the index into the general term. Each succeeding term of the summation is formed using successive integral values of the index, until the upper limit is reached.

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Aim: Summation Notation Course: Alg. 2 & Trig. Properties of Sums

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Aim: Summation Notation Course: Alg. 2 & Trig. Model Problems Find the value of each summation = 2(0 + 1 + 2 + 3 + 4 + 5 + 6) = 2(21) = 42 = (2 – 2) 2 + (3 – 2) 2 + (4 – 2) 2 + (5 – 2) 2 = 0 2 + 1 2 + 2 2 + 3 2 = 14

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Aim: Summation Notation Course: Alg. 2 & Trig. Model Problems Rewrite using the summation symbol 2(1) + 2(2) + 2(3) + 2(4) + 2(5)

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Aim: Summation Notation Course: Alg. 2 & Trig. Regents Questions = 2

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Aim: Summation Notation Course: Alg. 2 & Trig. In an arithmetic series, if a 1 is the first term, n is the number of terms, a n is the n th term, and d is the common difference, then S n the sum of the arithmetic series, is given by the formulas: The Sum of an Arithmetic Sequence: Series or

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Aim: Summation Notation Course: Alg. 2 & Trig. The n th Term of an Arithmetic Sequence The n th term of an arithmetic sequence has the form a n = dn + c where d is the common difference between consecutive terms of the sequence and c = a 1 – d An alternative form of the nth term is a n = a 1 + (n – 1)d Summation and Arithmetic Series

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Aim: Summation Notation Course: Alg. 2 & Trig. Regents Questions Find the partial sum of the following arithmetic series = 115

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Aim: Summation Notation Course: Alg. 2 & Trig. The Sum of a Finite Geometric Sequence The sum of the finite geometric sequence a 1, a 1 r 2, a 1 r 3, a 1 r 4,.... a 1 r n - 1.... with common ratio r 1 is given by Find the sum of the first eight terms of the geometric sequence 1, 3, 9, 27,...

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Aim: Summation Notation Course: Alg. 2 & Trig. Find the sum of the first eight terms of the geometric sequence 1, 3, 9, 27,... The Sum of a Finite Geometric Sequence r = ?3 a 1 a 2 a 3 a 4..... a n.... a 1 a 1 r a 1 r 2 a 1 r 3.... a 1 r n - 1.... a 1 + a 1 r + a 1 r 2 +a 1 r 3 +... + a 1 r 8 - 1 The sum of the finite geometric sequence =

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Aim: Summation Notation Course: Alg. 2 & Trig. Find the sum of the first eight terms of the geometric sequence 1, 3, 9, 27,... The Sum of a Finite Geometric Sequence r = ?3 1 + 3 + 9 + 27 +...+ 1(3) 7 = 3280 1 + 1(3) + 1(3) 2 + 1(3) 3 +....= 3280 =

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Aim: Summation Notation Course: Alg. 2 & Trig. Find the sum of Model Problem a 1 = 10; r = -1/2 How many terms in the series? starting at i = 0, there are 11 terms ; n = 11

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Aim: Summation Notation Course: Alg. 2 & Trig. Regents Questions Find the sum of the following geometric series. = 4368

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Aim: Summation Notation Course: Alg. 2 & Trig. Find = 4 + 4(0.6) + 4(0.6) 2 + 4(0.6) 3 +.. The Sum of a Infinite Geometric Sequence If |r| < 1, then the infinite geometric sequence a 1, a 1 r 2, a 1 r 3, a 1 r 4,.... a 1 r n - 1... has the sum a 1 = 4 and r = 0.6 - (|r| < 1)

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Aim: Summation Notation Course: Alg. 2 & Trig. Model Problem Find the sum of 3 + 0.3 + 0.03 + 0.003 +... a 1 = 3 and r = ?0.1 3 + 3(0.1) + 3(0.1) 2 + 3(0.1) 3 +...

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Aim: Summation Notation Course: Alg. 2 & Trig. Model Problem Express the series in sigma notation and find the sum. 54 + 18 + 6 + 2 + 2/3 + 2/9 728/9

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Aim: Summation Notation Course: Alg. 2 & Trig. Model Problem Find the sum of a 1 = 4(0.3) 1 = 1.2n = 12r = 0.3 1.714

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