Pre Calculus 11 Section 1.4 Geometric Series i) Sums of a geometric series and infinite geometric series ii) Deriving the formula for the sum of a geometric series and infinite geometric series © Copyright all rights reserved to Homework depot: www.BCMath.ca

Definition: Geometric Series A geometric series is the sum [addition] of the terms in a geometric sequence Consider the geometric sequence 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 . If we add the terms of the sequence, we can write the geometric series as In this section we will learn to find the value of a geometric series

II) Sum of Geometric Sequence: Sum of a geometric series up to the nth term Multiply both sides by “r” Subtract the equations! Factor out “Sn” Divide both sides by (1 - r) Factor out “a” © Copyright all rights reserved to Homework depot: www.BCMath.ca

Alternative Equation: Since the last term tn is: We can rewrite the equation as: The sum of a geometric series from the first term to the nth term First term (a) Common ratio times the last term One minus the common ratio Sum of the first 2 terms Sum of the first 5 terms

Ex: Find the sum of the following Geometric Sequence: © Copyright all rights reserved to Homework depot: www.BCMath.ca

Practice: Determine the sum of 14 terms of the geometric series: S = 6 + 18 + 54 + … ii) find the sum of the first 20 terms.

Example 4: Determine the nth term, and the sum of the first n terms of the geometric sequence which has 2, 6, and 18 as its first three numbers. The general term is an equation in terms of “n”. The value of the Geometric series varies as the value of “n” changes

Ex: Find the Sum: First find out how many terms there are Since “n” is a whole number we can guess and check Then find the sum up to the “14th”term © Copyright all rights reserved to Homework depot: www.BCMath.ca

Jason gave his son a penny on the first day of the month and doubled the amount each day. How much money will he give his son altogether by the 30th day? © Copyright all rights reserved to Homework depot: www.BCMath.ca

Example 5: The following is a school trip telephoning tree. Level 1: Teacher Level 2: Students Level 3: Students a) At what level are 64 students contacted? r = 2; (Each student contacts 2 students.) Find n .

b) How many are contacted at the 8th level? C) By the 8th level, how many students, in total, have been contacted? Remember that the teacher is not counted. So we will subtract 1 from the total S8 . The total number of students contacted is 255 – 1 = 254 .

d) By the nth level, how many students, in total, have been contacted? Again, remember that the teacher is not counted. So we will simply subtract 1 from the sum up to level n. The total number of students by level n is given by If there are 300 students in total, by what level will all have been contacted? Since 254 students have been contacted by level 8, then all 300 will have been telephoned by level 9.

Ex: A rubber ball is dropped from a height of 10m Ex: A rubber ball is dropped from a height of 10m. After each bounce, the ball returns to 65% of its previous height. Calculate all the vertical displacement right before the 8th bounce. Note: There are two types of displacement: Going up and down Each displacement is doubled except the first bounce © Copyright all rights reserved to Homework depot: www.BCMath.ca

Challenge: the sum of the second & Third term in a geometric series is 45. The sum of the fourth & Fifth term is 20. Find the geometric sequence: Use the common ratio to find the first term The 2nd and 3rd terms add to 45 The 4th and 5th terms add to 20 Factor out any common factors In each equation Divide the equations Find the 2nd series when the common ratio is negative © Copyright all rights reserved to Homework depot: www.BCMath.ca

Homework: Assignment 1.4