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Published byDerrick Gilbert Modified over 5 years ago

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Warm up 7, 15, 23, 31… Is this sequence arithmetic or geometric?

What is the recursive definition? What is the explicit formula? Arithmetic f(1) = 7 f(n) = f(n-1) + 8 f(n) = 7 + (n-1)8

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Today’s Objective: I can find the sum of an arithmetic series.

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The first four rows of chairs are set up for a meeting

The first four rows of chairs are set up for a meeting. The seating pattern is to continue through 20 rows. How many chairs will there be in all 20 rows? = 270 20 2 ( ) 𝑆 20 = 4 23 = 270 ··· + 100 50 2 ( ) Series: Sum of the terms of a sequence 𝑆 50 = 2 100 Finite Series: Has first and last term Infinite Series: Continues without end =2550 Sum of a Finite Arithmetic Series 𝑓(1)+𝑓(2)+𝑓(3)+⋯+𝑓(𝑛) 𝑆 𝑛 = 𝑛 2 (𝑓(1)+𝑓(𝑛))

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Summation Notation 𝑙𝑜𝑤𝑒𝑠𝑡 𝑡𝑒𝑟𝑚 𝑛𝑢𝑚𝑏𝑒𝑟 𝑢𝑝𝑝𝑒𝑟 𝑡𝑒𝑟𝑚 𝑛𝑢𝑚𝑏𝑒𝑟 𝐸𝑥𝑝𝑙𝑖𝑐𝑖𝑡 𝐹𝑜𝑟𝑚𝑢𝑙𝑎 – ··· 𝑛=1 40 7𝑛−12 ··· + 207 51 Arithmetic Series Explicit Formula = Linear Slope = Common Difference 4𝑛+3 Step 1 𝑛=1 𝐸𝑥𝑝𝑙𝑖𝑐𝑖𝑡 𝐹𝑜𝑟𝑚𝑢𝑙𝑎 𝑢𝑝𝑝𝑒𝑟 𝑡𝑒𝑟𝑚 𝑛𝑢𝑚𝑏𝑒𝑟 Step 2 𝑓(𝑛)= 7+(𝑛−1)⋅4 207=4𝑛+3 =7+4𝑛−4 204=4𝑛 =4𝑛+3 51=𝑛

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Find the sum of a Finite Arithmetic Series

𝑛=1 40 (3𝑛−8) 𝑛=1 4 𝑛 3 𝑓(1)= 1 3 =1 𝑓(2)= 2 3 =8 𝑆 𝑛 = 𝑛 2 (𝑓(1)+𝑓(𝑛)) 𝑓(3)= 3 3 =27 𝑓(4)= 4 3 =64 NOT linear!! Step 1: calculate the value of the first term 𝑓(1)= 3 1 −8 =−5 100 Step 2: calculate the value of the last term 𝑓(40)= 3 40 −8 =112 Step 3: calculate the sum of all 40 terms 40 2 (−5+112) 𝑆 40 = =𝟐𝟏𝟒𝟎

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A company pays a $10,000 bonus to salespeople at the end of their first 50 weeks if they make 10 sales in their first week, and then improve their sales numbers by two each week thereafter. One salesperson qualified for the bonus with the minimum possible number of sales. How many sales did the salesperson make in week 50? 𝑓(50)= 10+(50−1)⋅2 =108 How many sales did the salesperson make in all 50 weeks? 50 2 (10+108) p. 591: 9-45 mult. of 3 and 47 𝑆 50 = =2950

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