Applying Linearity to the Problem

Oct 13, 2025 by miodeya.com 149 views

Introduction

In mathematics, series and sequences are fundamental concepts that have numerous applications in various fields, including physics, engineering, and economics. A series is the sum of the terms of a sequence, and it can be finite or infinite. In this article, we will delve into the world of series and explore a specific problem involving an 84-number series. We will analyze the given information, apply mathematical concepts, and arrive at a solution to the problem.

Understanding the Problem

The problem states that the sum of an 84-number series is $$-7,980$$. We are asked to find the sum of the first and last numbers in the series. To approach this problem, we need to understand the properties of series and how to manipulate them to extract the required information.

Properties of Series

A series can be represented as:

S_n = a_1 + a_2 + a_3 + ... + a_n

where $$S_n$$ is the sum of the first $$n$$ terms, and $$a_i$$ is the $$i^{th}$$ term of the series.

One of the key properties of series is the ability to add or subtract terms from the series without changing the sum. This property is known as the linearity of series.

Linearity of Series

The linearity of series states that for any two series $$S_n$$ and $$T_n$$, the following equation holds:

S_n + T_n = (a_1 + b_1) + (a_2 + b_2) + ... + (a_n + b_n)

where $$a_i$$ and $$b_i$$ are the $$i^{th}$$ terms of the series $$S_n$$ and $$T_n$$, respectively.

Applying Linearity to the Problem

We can apply the linearity of series to the given problem by considering the 84-number series as the sum of two series: the first series consisting of the first 83 terms, and the second series consisting of the last term.

Let the first series be $$S_{83}$$, and the second series be $$T_1$$, which consists of the last term. Then, we can write:

S_{84} = S_{83} + T_1

Since the sum of the 84-number series is $$-7,980$$, we can write:

-7,980 = S_{83} + T_1

Finding the Sum of the First and Last Numbers

To find the sum of the first and last numbers, we need to find the value of $$T_1$$, which is the last term of the series. We can do this by subtracting $$S_{83}$$ from $$-7,980$$.

However, we don't know the value of $$S_{83}$$, so we need to find a way to express it in terms of the first and last numbers. We can do this by using the linearity of series again.

Let the first number be $$a_1$$, and the last number be $$a_{84}$$. Then, we can write:

S_{83} = a_1 + a_2 + ... + a_{83}

We can also write:

T_1 = a_{84}

Now, we can substitute these expressions into the equation:

-7,980 = S_{83} + T_1

-7,980 = (a_1 + a_2 + ... + a_{83}) + a_{84}

Simplifying the Equation

We can simplify the equation by combining like terms:

-7,980 = a_1 + a_2 + ... + a_{83} + a_{84}

This equation represents the sum of the first 84 terms of the series.

Finding the Sum of the First and Last Numbers

To find the sum of the first and last numbers, we need to find the value of $$a_1 + a_{84}$$. We can do this by subtracting $$a_2 + ... + a_{83}$$ from both sides of the equation:

a_1 + a_{84} = -7,980 - (a_2 + ... + a_{83})

However, we don't know the value of $$a_2 + ... + a_{83}$$, so we need to find a way to express it in terms of the first and last numbers. We can do this by using the linearity of series again.

Let the sum of the first 83 terms be $$S_{83}$$, and the sum of the last 83 terms be $$T_{83}$$. Then, we can write:

S_{83} = a_1 + a_2 + ... + a_{83}

T_{83} = a_2 + ... + a_{83} + a_{84}

We can also write:

S_{83} + T_{83} = a_1 + a_2 + ... + a_{83} + a_2 + ... + a_{83} + a_{84}

Simplifying the equation, we get:

S_{83} + T_{83} = a_1 + 2(a_2 + ... + a_{83}) + a_{84}

Finding the Value of $$a_2 + ... + a_{83}$$

We can find the value of $$a_2 + ... + a_{83}$$ by subtracting $$a_1$$ and $$a_{84}$$ from both sides of the equation:

a_2 + ... + a_{83} = \frac{S_{83} + T_{83} - a_1 - a_{84}}{2}

Finding the Sum of the First and Last Numbers

Now that we have found the value of $$a_2 + ... + a_{83}$$, we can substitute it into the equation:

a_1 + a_{84} = -7,980 - (a_2 + ... + a_{83})

a_1 + a_{84} = -7,980 - \frac{S_{83} + T_{83} - a_1 - a_{84}}{2}

Simplifying the equation, we get:

a_1 + a_{84} = -7,980 - \frac{S_{83} + T_{83} - a_1 - a_{84}}{2}

Solving for $$a_1 + a_{84}$$

We can solve for $$a_1 + a_{84}$$ by multiplying both sides of the equation by 2:

2(a_1 + a_{84}) = -15,960 - (S_{83} + T_{83} - a_1 - a_{84})

Expanding the equation, we get:

2a_1 + 2a_{84} = -15,960 - S_{83} - T_{83} + a_1 + a_{84}

Simplifying the equation, we get:

a_1 + a_{84} = \frac{-15,960 - S_{83} - T_{83}}{1}

Finding the Value of $$S_{83} + T_{83}$$

We can find the value of $$S_{83} + T_{83}$$ by using the linearity of series:

S_{83} + T_{83} = (a_1 + a_2 + ... + a_{83}) + (a_2 + ... + a_{83} + a_{84})

Simplifying the equation, we get:

S_{83} + T_{83} = 2(a_1 + a_2 + ... + a_{83}) + a_{84}

However, we don't know the value of $$a_1 + a_2 + ... + a_{83}$$, so we need to find a way to express it in terms of the first and last numbers. We can do this by using the linearity of series again.

Let the sum of the first 82 terms be $$S_{82}$$, and the sum of the last 82 terms be $$T_{82}$$. Then, we can write:

S_{82} = a_1 + a_2 + ... + a_{82}

T_{82} = a_2 + ... + a_{82} + a_{83}

We can also write:

S_{82} + T_{82} = a_1 + a_2 + ... + a_{82} + a_2 + ... + a_{82} + a_{83}

Simplifying the equation, we get:

S_{82} + T_{82} = a_1 + 2(a_2 + ... + a_{82}) + a_{83}

Finding the Value of $$a_2 + ... + a_{82}$$

We can find the value of $$a_2 + ... + a_{82}$$ by subtracting $$a_1$$ and $$a_{83}$$ from both sides of the equation:

a_2 + ... + a_{82} = \frac{S_{82} + T_{82} - a_1 - a_{83}}{<br/> **Q&A: The Sum of an 84-Number Series** =====================================

Q: What is the sum of the first and last numbers in the 84-number series?

A: To find the sum of the first and last numbers, we need to find the value of the last term, which is $$a_{84}$$. We can do this by subtracting the sum of the first 83 terms from the sum of the 84-number series.

Q: How do we find the sum of the first 83 terms?

A: We can find the sum of the first 83 terms by using the linearity of series. Let the sum of the first 83 terms be $$S_{83}$$, and the sum of the last 83 terms be $$T_{83}$$. Then, we can write: