Understanding the Quadratic Sequence

Oct 13, 2025 by windailys.com 137 views

Solving a Quadratic Sequence: Finding the Values of a and b

In mathematics, a sequence is a series of numbers in a specific order. Quadratic sequences are a type of sequence where each term is defined by a quadratic expression. In this article, we will explore a quadratic sequence defined by the formula ${$ r_n = a n^2 + b n $}$, where ${$ a, b \in \mathbb{R} $}$. We will use the given conditions ${$ T_1 = 7 $}$ and ${$ T_2 = 20 $}$ to find the values of ${$ a $}$ and ${$ b $}$.

Understanding the Quadratic Sequence

A quadratic sequence is a sequence where each term is defined by a quadratic expression. In this case, the quadratic expression is ${$ a n^2 + b n $}$. This means that each term in the sequence is a quadratic function of the term number ${$ n $}$.

Given Conditions

We are given two conditions: ${$ T_1 = 7 $}$ and ${$ T_2 = 20 $}$. These conditions mean that the first term ${$ T_1 $}$ is equal to 7, and the second term ${$ T_2 $}$ is equal to 20.

Finding the Values of a and b

To find the values of ${$ a $}$ and ${$ b $}$, we can use the given conditions to set up a system of equations. We know that ${$ T_1 = r_1 = a (1)^2 + b (1) = a + b $}$ and ${$ T_2 = r_2 = a (2)^2 + b (2) = 4a + 2b $}$.

We can now set up a system of equations using these two expressions:

${$ a + b = 7 $}{$ 4a + 2b = 20 $}$

Solving the System of Equations

To solve the system of equations, we can use the method of substitution or elimination. Let's use the elimination method.

First, we can multiply the first equation by 2 to get:

${$ 2a + 2b = 14 $}$

Now, we can subtract this equation from the second equation to get:

${$ 2a + 2b - (2a + 2b) = 20 - 14 $}{$ 0 = 6 $}$

This is a contradiction, which means that the system of equations has no solution. However, we can try to find a solution by using the method of substitution.

Let's solve the first equation for ${$ a $}$:

${$ a = 7 - b $}$

Now, we can substitute this expression for ${$ a $}$ into the second equation:

${$ 4(7 - b) + 2b = 20 $}{$ 28 - 4b + 2b = 20 $}{$ 28 - 2b = 20 $}{$ -2b = -8 $}{$ b = 4 $}$

Now that we have found the value of ${$ b $}$, we can find the value of ${$ a $}$:

${$ a = 7 - b $}{$ a = 7 - 4 $}{$ a = 3 $}$

In this article, we have explored a quadratic sequence defined by the formula ${$ r_n = a n^2 + b n $}$, where ${$ a, b \in \mathbb{R} $}$. We have used the given conditions ${$ T_1 = 7 $}$ and ${$ T_2 = 20 $}$ to find the values of ${$ a $}$ and ${$ b $}$. We have shown that the values of ${$ a $}$ and ${$ b $}$ are ${$ a = 3 $}$ and ${$ b = 4 $}$.

In our previous article, we explored a quadratic sequence defined by the formula ${$ r_n = a n^2 + b n $}$, where ${$ a, b \in \mathbb{R} $}$. We used the given conditions ${$ T_1 = 7 $}$ and ${$ T_2 = 20 $}$ to find the values of ${$ a $}$ and ${$ b $}$. In this article, we will answer some frequently asked questions about quadratic sequences.

Q: What is a quadratic sequence?

A: A quadratic sequence is a sequence where each term is defined by a quadratic expression. In this case, the quadratic expression is ${$ a n^2 + b n $}$. This means that each term in the sequence is a quadratic function of the term number ${$ n $}$.

Q: How do I find the values of a and b in a quadratic sequence?

A: To find the values of ${$ a $}$ and ${$ b $}$, you can use the given conditions to set up a system of equations. You can then use the method of substitution or elimination to solve the system of equations.

Q: What if the system of equations has no solution?

A: If the system of equations has no solution, it means that the given conditions are inconsistent. In this case, you may need to re-examine the given conditions or use a different method to find the values of ${$ a $}$ and ${$ b $}$.

Q: Can I use the quadratic formula to find the values of a and b?

A: No, the quadratic formula is used to find the roots of a quadratic equation, not the values of ${$ a $}$ and ${$ b $}$ in a quadratic sequence.

Q: How do I determine the nth term of a quadratic sequence?

A: To determine the nth term of a quadratic sequence, you can use the formula ${$ r_n = a n^2 + b n $}$. Simply substitute the value of ${$ n $}$ into the formula to find the nth term.

Q: Can I use a quadratic sequence to model real-world phenomena?

A: Yes, quadratic sequences can be used to model many real-world phenomena, such as the motion of an object under the influence of gravity, the growth of a population, or the cost of a product over time.

Q: What are some common applications of quadratic sequences?

A: Quadratic sequences have many applications in fields such as physics, engineering, economics, and computer science. Some common applications include:

Modeling the motion of an object under the influence of gravity
Modeling the growth of a population
Modeling the cost of a product over time
Modeling the behavior of a system with quadratic feedback

In this article, we have answered some frequently asked questions about quadratic sequences. We have shown that quadratic sequences are a powerful tool for modeling real-world phenomena and have many applications in fields such as physics, engineering, economics, and computer science.

The final answer is: $\boxed{Quadratic sequences are a powerful tool for modeling real-world phenomena and have many applications in fields such as physics, engineering, economics, and computer science.}$