The Sum Of The First 5 Terms Of A Convergent Geometric Series Is 62, And The Sum To Infinity Of The Series Is 64. Determine The Common Ratio.

Mar 12, 2025 by ADMIN 142 views

**The Sum of the First 5 Terms of a Convergent Geometric Series**

Introduction

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The sum of the first n terms of a geometric series can be calculated using the formula: S_n = a * (1 - r^n) / (1 - r), where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms. In this article, we will discuss how to determine the common ratio of a convergent geometric series given the sum of the first 5 terms and the sum to infinity.

The Sum of the First 5 Terms

The sum of the first 5 terms of a geometric series is given as 62. We can use the formula for the sum of the first n terms to write an equation:

a * (1 - r^5) / (1 - r) = 62

The Sum to Infinity

The sum to infinity of a geometric series is given as 64. We can use the formula for the sum to infinity to write an equation:

a / (1 - r) = 64

Solving for the Common Ratio

We can solve for the common ratio by first solving the second equation for a:

a = 64 * (1 - r)

Now, substitute this expression for a into the first equation:

64 * (1 - r) * (1 - r^5) / (1 - r) = 62

Simplify the equation:

64 * (1 - r^5) = 62 * (1 - r)

Expand the equation:

64 - 64r^5 = 62 - 62r

Rearrange the equation:

64r^5 - 62r = 2

Factor out r:

r(64r^4 - 62) = 2

Divide both sides by 2:

r(32r^4 - 31) = 1

Divide both sides by (32r^4 - 31):

r = 1 / (32r^4 - 31)

Solving for the Common Ratio (continued)

To solve for the common ratio, we need to find a value of r that satisfies the equation. We can start by noticing that the equation is a quartic equation in r. However, we can simplify the equation by noticing that r = 1 is a solution. We can factor out (r - 1) from the equation:

r(32r^4 - 31) = 1

(r - 1)(32r^4 + 32r^3 + 32r^2 + 32r - 31) = 0

Since r = 1 is a solution, we can divide both sides by (r - 1):

32r^4 + 32r^3 + 32r^2 + 32r - 31 = 0

Solving the Quartic Equation

To solve the quartic equation, we can try to factor it or use a numerical method. However, in this case, we can use a clever trick to solve the equation. Notice that the equation can be written as:

(8r^2 + 8r + 8)(4r^2 - 4r + 4) - 31 = 0

Expand the equation:

32r^4 + 32r^3 + 32r^2 + 32r - 31 = 0

Simplify the equation:

(8r^2 + 8r + 8)(4r^2 - 4r + 4) = 31

Now, notice that the left-hand side of the equation is a product of two quadratic expressions. We can try to factor the quadratic expressions:

(8r^2 + 8r + 8) = (2r + 1)(4r + 4)

(4r^2 - 4r + 4) = (2r - 1)(2r + 2)

Now, substitute these factorizations back into the equation:

(2r + 1)(4r + 4)(2r - 1)(2r + 2) = 31

Expand the equation:

(4r^2 + 8r + 4)(4r^2 - 4r + 4) = 31

Simplify the equation:

16r^4 + 16r^3 - 16r^2 + 16r + 16 = 31

Subtract 31 from both sides:

16r^4 + 16r^3 - 16r^2 + 16r - 15 = 0

Solving the Quartic Equation (continued)

To solve the quartic equation, we can try to factor it or use a numerical method. However, in this case, we can use a clever trick to solve the equation. Notice that the equation can be written as:

(4r^2 + 4r - 5)(4r^2 - 4r + 3) = 0

Expand the equation:

16r^4 + 16r^3 - 16r^2 + 16r - 15 = 0

Simplify the equation:

(4r^2 + 4r - 5)(4r^2 - 4r + 3) = 0

Now, notice that the left-hand side of the equation is a product of two quadratic expressions. We can try to factor the quadratic expressions:

(4r^2 + 4r - 5) = (2r + 5)(2r - 1)

(4r^2 - 4r + 3) = (2r - 1)(2r + 1)

Now, substitute these factorizations back into the equation:

(2r + 5)(2r - 1)(2r - 1)(2r + 1) = 0

Expand the equation:

(2r + 5)(2r - 1)^2(2r + 1) = 0

Simplify the equation:

(2r + 5)(4r^2 - 4r + 1) = 0

Expand the equation:

8r^3 + 12r^2 - 4r - 5 = 0

Solving the Cubic Equation

To solve the cubic equation, we can try to factor it or use a numerical method. However, in this case, we can use a clever trick to solve the equation. Notice that the equation can be written as:

(2r + 5)(4r^2 - 4r + 1) = 0

Expand the equation:

8r^3 + 12r^2 - 4r - 5 = 0

Simplify the equation:

(2r + 5)(4r^2 - 4r + 1) = 0

Now, notice that the left-hand side of the equation is a product of two expressions. We can try to factor the expressions:

(2r + 5) = 0

(4r^2 - 4r + 1) = 0

Solve the first equation:

2r + 5 = 0

Subtract 5 from both sides:

2r = -5

Divide both sides by 2:

r = -5/2

Solve the second equation:

4r^2 - 4r + 1 = 0

Expand the equation:

4r^2 - 4r + 1 = 0

Simplify the equation:

(2r - 1)^2 = 0

Take the square root of both sides:

2r - 1 = 0

Add 1 to both sides:

2r = 1

Divide both sides by 2:

r = 1/2

Conclusion

Q: What is a convergent geometric series?

A: A convergent geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The series converges to a finite sum if the absolute value of the common ratio is less than 1.

Q: How do I determine the common ratio of a convergent geometric series?

A: To determine the common ratio of a convergent geometric series, you need to know the sum of the first n terms and the sum to infinity. You can use the formula for the sum of the first n terms and the sum to infinity to write two equations. Then, you can solve the equations to find the common ratio.

Q: What are the formulas for the sum of the first n terms and the sum to infinity of a geometric series?

A: The formula for the sum of the first n terms of a geometric series is:

S_n = a * (1 - r^n) / (1 - r)

where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.

The formula for the sum to infinity of a geometric series is:

S = a / (1 - r)

where S is the sum to infinity, a is the first term, and r is the common ratio.

Q: How do I use the formulas to determine the common ratio?

A: To use the formulas to determine the common ratio, you need to know the sum of the first n terms and the sum to infinity. You can substitute these values into the formulas and solve for the common ratio.

Q: What if I have a quadratic equation in the common ratio? How do I solve it?

A: If you have a quadratic equation in the common ratio, you can try to factor it or use a numerical method to solve it. Alternatively, you can use the quadratic formula to solve for the common ratio.

Q: What if I have a cubic equation in the common ratio? How do I solve it?

A: If you have a cubic equation in the common ratio, you can try to factor it or use a numerical method to solve it. Alternatively, you can use a cubic formula to solve for the common ratio.

Q: What if I have a quartic equation in the common ratio? How do I solve it?

A: If you have a quartic equation in the common ratio, you can try to factor it or use a numerical method to solve it. Alternatively, you can use a quartic formula to solve for the common ratio.

Q: What are some common mistakes to avoid when determining the common ratio?

A: Some common mistakes to avoid when determining the common ratio include:

Not checking if the series converges before trying to determine the common ratio
Not using the correct formulas for the sum of the first n terms and the sum to infinity
Not solving the equations correctly
Not checking if the solutions are valid (i.e., the common ratio is not equal to 1)

Q: How do I check if the series converges?

A: To check if the series converges, you need to check if the absolute value of the common ratio is less than 1. If it is, then the series converges. If it is not, then the series diverges.

Q: What if I have a complex common ratio? How do I handle it?

A: If you have a complex common ratio, you can handle it by using the complex conjugate of the common ratio to simplify the calculations. Alternatively, you can use a numerical method to solve for the common ratio.

Q: What are some real-world applications of determining the common ratio?

A: Some real-world applications of determining the common ratio include:

Finance: Determining the common ratio of a geometric series can help investors understand the growth rate of their investments.
Economics: Determining the common ratio of a geometric series can help economists understand the growth rate of economic indicators such as GDP.
Science: Determining the common ratio of a geometric series can help scientists understand the growth rate of physical phenomena such as population growth.

Conclusion

In this article, we discussed how to determine the common ratio of a convergent geometric series given the sum of the first n terms and the sum to infinity. We also discussed some common mistakes to avoid and some real-world applications of determining the common ratio.