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Here are a few recommended readings before getting started with this lesson.
The following applet shows the first five terms of a sum. Identify whether the given sum is a geometric series or not.
Let a1 and r be the first term and the common ratio, respectively, of a geometric sequence with n terms, where r=1. The sum of the related finite geometric series can be found by using the following formula.
Sn=1−ra1(1−rn), r=1
LHS⋅(1−r)=RHS⋅(1−r)
Distribute (1−r)
Multiply
Add and subtract terms
LHS/(1−r)=RHS/(1−r)
Sn=1−ra1(1−rn)
Substitute values
Identity Property of Multiplication
ba=b⋅(-1)a⋅(-1)
Calculate power
Subtract terms
Calculate quotient
As the number of infected students increased, the school decided to start a quarantine period. During the quarantine period at North High School, lessons started being taught online. In one of his math lessons, Tearrik's math teacher introduced a geometric series an example written using sigma notation.
Calculate the given sum.Use the formula for the sum of a finite geometric series.
Calculate the sum of all the terms of the given finite geometric series written in summation notation. Remember that the formula for the sum of a geometric series can be used rather than adding the terms one by one.
Let a1 and r be the first term and the common ratio, respectively, of a geometric sequence with n terms, where r=1. For an infinite series, if the common ratio r is greater than -1 and less than 1 — in other words, if ∣r∣<1 — then the sum can be found by using the following formula.
S∞=1−ra1, -1<r<1
This means that the sum converges on a number. If the common ratio r is less than or equal to -1 or greater than or equal to 1 — if ∣r∣≥1 — then the sum diverges. In such cases, there is no sum for the infinite geometric series.
rn=0, n=∞
Subtract term
Identity Property of Multiplication
S∞=1−ra1
Sum: 3
Sum: No sum.
a1=1, r=32
Rewrite 1 as 33
Subtract fractions
a/b1=ab
Simplify quotient
a1=-410, a2=1650
c/da/b=ba⋅cd
Put minus sign in numerator
Multiply fractions
ba=b/40a/40
Put minus sign in front of fraction
Determine whether the given infinite geometric series converge or diverge. Remember that if the common ratio ∣r∣<1, then the infinite series converges to a number, and that if ∣r∣≥1, then the series diverges.
If the common ratio of an infinite geometric series is less than or equal to -1 or greater than or equal to 1, the sum of the series does not exist. However, it is possible to find a partial sum or the sum of the first several terms in the series. This partial series can be thought of as a finite series. As such, its sum can be found using the formula for a finite geometric series.
Sn=1−ra1(1−rn), r=1
After recovering from his illness, Tearrik returns to school and continues to play basketball with his best friend Tadeo. Suppose that after the ball hits the rim of the basket, the ball falls 3 meters and rebounds to 85% of the height of the previous bounce.
Calculate power
Multiply
Add terms
Round to 2 decimal place(s)
Now that they are completely recovered, Tearrik and Tadeo decide to save money so they can go to the NBA finals next year, 15 months from now. They start chatting about their own ways to save money.
Who will save more money in 15 months?Recall that a series is geometric if it has a common ratio.
To determine who will save more money in 15 months, the total amounts saved will be calculated one at a time.
Tearrik will start by saving only $0.50 the first month. Then, he will increase the amount of money he saves every month by doubling the previous amount.
Since there is a common ratio between the amount of money saved each month, this sum represents a geometric series. Therefore, the formula for the sum of a finite geometric series will be used to calculate the total amount of money Tearrik will have saved in 15 months.