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Chapter 9-3 Arithmetic Sequences and Series (Day 2) Obj: To find the sums of an Arithmetic Series. Who uses this? – You can use arithmetic sequences to predict costs. See number 36 in homework Consider the Sum of the following arithmetic sequence 1, 2, 3, 4, 5, 6, 7 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 The sum of the terms of a sequence is called a series. The series above is an Arithmetic Series. (Why…?) Arithmetic Sequence vs. Arithmetic Series 4, 7, 10, 13, 16 4 + 7 + 10 + 13 + 16 Sum of an Arithmetic Series - The Sum (๐๐ )of the first ๐ terms is: ๐ ๐๐ = (๐1 + ๐๐ ) 2 Sum # of Terms 1๐ ๐ก Term ๐๐กโ Term (end) ๐ *Use your Calculator and the formula ๐ ๐ = (๐1 + ๐๐ ) 2 Ex 1. Find the sum of the first 50 positive even integers ๐ ๐๐ = 2 (๐1 + ๐๐ ) ๐50 = Sub in the values for ๐, ๐1, & ๐๐ 50 (2 + 100) 2 ๐50 = 25(102) = 2250 Ex 2. Find the sum of the whole numbers from 1 to 100 What if we are looking for the sum of the following Arithmetic Series? ๐ ๐40 ๐๐๐ 70 + 49 + 28 + 7 + โฏ ๐๐ = (๐1 + ๐๐ ) 2 Step 1 – find ๐๐ using ๐๐ = ๐1 + (๐ − 1)๐ Step 2 – plug the values for ๐๐, ๐, ๐1 OR… ๐ & ๐๐ into ๐๐ = (๐1 + ๐๐ ) 2 ๐ ๐๐ = (๐1 + ๐๐ ) 2 ๐๐ = ๐1 + (๐ − 1) Sub in ๐ ๐๐ = {๐1 + [๐1 + (๐ − 1)]} 2 which simplifies to.. ๐ ๐๐ = 2 [2๐1 + (๐ − 1)๐] **This formula can be used when you do not know the value of the last term. Now find the same sum using the above formula ๐40 ๐๐๐ 70 + 49 + 28 + 7 + โฏ You try … Find ๐15 ๐๐๐ 25 + 12 + (−1) + (−14) + โฏ Sigma Notation - (or Summation Notation) 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 = Can be expressed as: It is read “the sum of 2n as n increases from 1 to 10” Ex 1. Find Need to find the 1st and 12th terms Homework Ch 9-3 (day 2) pg 648, 17-19, 33-36