## Geometric Series Common Core Algebra 2 Homework Answers

If you are looking for geometric series common core algebra 2 homework answers, you have come to the right place. In this article, we will explain what geometric series are, how to find their general term and sum, and how to apply them to real-world problems. We will also provide some examples and exercises for you to practice your skills.

## Geometric Series Common Core Algebra 2 Homework Answers

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## What is a Geometric Series?

A geometric series is the sum of the terms of a geometric sequence. A geometric sequence is a sequence of numbers where each successive number is the product of the previous number and some constant r. The constant r is called the common ratio of the sequence. For example, the following is a geometric sequence with a common ratio of 3:

3, 9, 27, 81, 243, ...

The sum of the first n terms of a geometric sequence is called the n partial sum of the series. For example, the sum of the first 5 terms of the above sequence is:

S5 = 3 + 9 + 27 + 81 + 243 = 363

## How to Find the General Term and Sum of a Geometric Series?

To find the general term and sum of a geometric series, we need to know two things: the first term a1 and the common ratio r of the sequence. The general term an of a geometric sequence can be written as:

an = a1r

This formula tells us how to find any term of the sequence given its position n. For example, using the sequence above, we can find the 10 term as follows:

a10 = a1r = 3(3) = 19683

The sum Sn of the first n terms of a geometric series can be found using this formula:

Sn = a1(1 - r) / (1 - r)

This formula tells us how to find the partial sum of any number of terms given the first term and the common ratio. For example, using the sequence above, we can find the sum of the first 10 terms as follows:

S10 = a1(1 - r) / (1 - r) = 3(1 - (3)) / (1 - 3) = -98415 / -2 = 49207.5

## How to Apply Geometric Series to Real-World Problems?

Geometric series can be used to model many real-world phenomena that involve exponential growth or decay, such as population growth, compound interest, radioactive decay, etc. Here are some examples:

A bacteria culture starts with 1000 bacteria and doubles every hour. How many bacteria will there be after 8 hours?

A bank offers an annual interest rate of 5% compounded monthly. How much money will you have after depositing $1000 for 5 years?

A radioactive substance has a half-life of 20 days. How much of it will remain after 60 days if you start with 100 grams?

### Example 3: Radioactive Decay

Let An be the amount of radioactive substance after n days. Since the substance has a half-life of 20 days, we have a geometric sequence with a common ratio of r = 0.5. The first term is A1 = 100. Using the general term formula, we can write:

An = A1r = 100(0.5)

To find the amount of substance after 60 days, we substitute n = 60 into the formula:

A60 = 100(0.5) = 100(0.5) = 12.5

Therefore, there will be 12.5 grams of substance after 60 days.

## How to Find Geometric Series Common Core Algebra 2 Homework Answers?

If you are looking for geometric series common core algebra 2 homework answers, you can use the formulas and methods we have explained above to solve any geometric series problem. You can also check your answers using online calculators or websites that provide step-by-step solutions and explanations for algebra 2 problems. Here are some examples:

Mathway Algebra Calculator: This website allows you to enter any algebra problem and get instant solutions and explanations.

Symbolab Algebra Calculator: This website also provides solutions and explanations for algebra problems, as well as practice problems and quizzes.

Quizlet Algebra 2 Common Core Textbook Solutions: This website provides solutions and answers for the textbook Algebra 2: A Common Core Curriculum by Boswell and Larson.

### Example 4: Finding Sums of Infinite Geometric Series

Sometimes, a geometric series has infinitely many terms. For example, the following is an infinite geometric series with a common ratio of r = 0.5:

1 + 0.5 + 0.25 + 0.125 + ...

To find the sum of an infinite geometric series, we need to know the first term a1 and the common ratio r of the sequence. The sum S of an infinite geometric series can be found using this formula:

S = a1 / (1 - r)

This formula only works if the common ratio r is between -1 and 1, otherwise the series does not converge to a finite value. For example, using the series above, we can find the sum as follows:

S = a1 / (1 - r) = 1 / (1 - 0.5) = 1 / 0.5 = 2

Therefore, the sum of the infinite geometric series is 2.

### Example 5: Applying Infinite Geometric Series to Real-World Problems

Infinite geometric series can also be used to model some real-world situations that involve repeated fractions or decimals. Here are some examples:

A ball is dropped from a height of 10 feet and bounces back up to half its previous height each time. How far does the ball travel in total?

A person deposits $1000 into a bank account that pays 6% annual interest compounded continuously. How much money will the person have after 10 years?

A dartboard has a circular target with a radius of 10 inches. The target is divided into concentric rings, each with half the radius of the previous one. The outermost ring has a score of 1 point, and each inner ring has twice the score of the previous one. What is the expected score of a dart thrown at random on the target?

### Example 6: Ball Bounce

Let dn be the distance the ball travels in the n bounce. Since the ball bounces back up to half its previous height each time, we have a geometric sequence with a common ratio of r = 0.5. The first term is d1 = 10, which is the initial height of the ball. Using the general term formula, we can write:

dn = d1r = 10(0.5)

To find the total distance the ball travels, we need to add up all the terms of the sequence. However, since the sequence has infinitely many terms, we need to use the formula for the sum of an infinite geometric series. The sum S of the distance is:

S = d1 / (1 - r) = 10 / (1 - 0.5) = 10 / 0.5 = 20

Therefore, the ball travels 20 feet in total.

### Example 7: Continuous Compound Interest

Let At be the amount of money after t years. Since the interest is compounded continuously, we have an exponential function with a base of e. The initial amount is A0 = 1000 and the annual interest rate is r = 0.06. Using the formula for continuous compound interest, we can write:

At = A0e = 1000e

To find the amount of money after 10 years, we substitute t = 10 into the formula:

A10 = 1000e = 1000e 1822.12

Therefore, the person will have $1822.12 after 10 years.

### Example 8: Dartboard Score

Let S be the score of a dart thrown at random on the target. Since the target is divided into concentric rings, each with half the radius of the previous one, we have a geometric probability distribution with a common ratio of r = 0.5. The outermost ring has a score of s1 = 1 and each inner ring has twice the score of the previous one. Using the formula for geometric probability, we can write:

P(S = sn) = r(1 - r) = (0.5)(1 - 0.5) = (0.5)

To find the expected score of a dart thrown at random on the target, we need to multiply each possible score by its probability and add them up. However, since there are infinitely many possible scores, we need to use an infinite series to calculate the expected value E(S). The expected score E(S) is:

E(S) = s1P(S = s1) + s2P(S = s2) + s3P(S = s3) + ...

= (1)(0.5)+ (2)(0.5)+ (4)(0.5)+ ...

= (0.5 + 0.5 + 0.5 + ...)

= (1 / (1 - 0.5))

= 2

Therefore, the expected score of a dart thrown at random on the target is 2 points.

### Example 9: Finding the nth Term of a Geometric Sequence

Sometimes, we are given some terms of a geometric sequence and we need to find the nth term or the general term of the sequence. To do this, we need to find the first term a1 and the common ratio r of the sequence. We can use the general term formula to write a system of equations and solve for a1 and r. For example, suppose we are given the following terms of a geometric sequence:

a2 = 12 and a5 = 1.5

To find the nth term of the sequence, we write two equations using the general term formula:

a2 = a1r = a1r

a5 = a1r = a1r

We can divide the second equation by the first equation to eliminate a1 and solve for r:

(a5) / (a2) = (a1r) / (a1r) = r

(1.5) / (12) = r

r = 0.125

r = 0.5

We can substitute r = 0.5 into the first equation and solve for a1:

a2 = a1r

(12) = a1(0.5)

a1 = 24

We can substitute a1 = 24 and r = 0.5 into the general term formula and write the nth term of the sequence:

an = a1r

an= 24(0.5)

### Example 10: Finding the Sum of a Finite Geometric Series

Sometimes, we are given some terms of a geometric series and we need to find the sum of a finite number of terms of the series. To do this, we need to find the first term a1, the common ratio r, and the number of terms n of the series. We can use the general term formula and the sum formula to write a system of equations and solve for a1, r, and n. For example, suppose we are given the following terms of a geometric series:

S = 8 + 4 + 2 + ... + 0.125

To find the sum of this series, we write three equations using the general term formula and the sum formula:

an= 0.125

an+1 = 0

S = (an+1 - an ) / (r - 1)

We can use the second equation to solve for n:

an+1 = an r

(0) = (0.125)r

r = 0

This means that n is infinite, so we cannot use this equation to find n.

We can use the first equation to solve for r:

an = an-1 r </p

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## Conclusion

In this article, we have learned about geometric series and how to find their general term and sum. We have also seen how to apply geometric series to real-world problems that involve exponential growth or decay, compound interest, radioactive decay, etc. We have also learned how to find the nth term or the sum of a geometric series given some terms of the series. We hope this article has helped you understand and master geometric series common core algebra 2 homework answers. d282676c82

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