Problem 27 - Calculus

Question

Involve numbers \(\int_{0}, f_{1}, f_{2}, \ldots\) and their differences \(v_{j}=f_{j}-f_{j-1}\), They give practice with subscripts \(0, \ldots, j\). If \(f_{j}=j^{2}+j+1\) find a formula for \(v_{j}\).

Step-by-Step Solution

Answer

The formula for \( v_j \) is \( v_j = 2j \).

Step 1: Express the formula for sequences

The given sequence formula is \( f_j = j^2 + j + 1 \). We need to find a formula for \( v_j \), which represents the difference between consecutive terms \( f_j \) and \( f_{j-1} \).

Step 2: Determine the formula for previous term

Calculate \( f_{j-1} \) using the sequence formula by replacing \( j \) with \( j-1 \):\[ f_{j-1} = (j-1)^2 + (j-1) + 1 \] Simplifying this gives:\[ f_{j-1} = (j^2 - 2j + 1) + (j - 1) + 1 = j^2 - j + 1 \]

Step 3: Calculate differences between consecutive terms

Utilizing the definition \( v_j = f_j - f_{j-1} \), substitute \( f_j \) and \( f_{j-1} \):\[ v_j = (j^2 + j + 1) - (j^2 - j + 1) \]

Step 4: Simplify the expression

Simplify the expression obtained in Step 3:\[ v_j = j^2 + j + 1 - j^2 + j - 1 \]By combining like terms, we find that:\[ v_j = 2j \]

Step 5: Final Step: State the formula for the differences

The formula for the differences \( v_j \) is \( v_j = 2j \). This concludes the solution process, expressing \( v_j \) in terms of \( j \).

Key Concepts

Sequences

In discrete mathematics, a sequence is an ordered list of numbers where each element is called a term. These terms are indexed using natural numbers, starting from 0 or 1, depending on the context. Sequences can be finite or infinite and are denoted by a function that assigns a number to each integer.

A sequence can often be expressed with a formula, like the one in our example:

• The given sequence is defined as \( f_j = j^2 + j + 1 \).

This means for each integer value of \( j \), there is a corresponding term in the sequence.

Understanding sequences is crucial because they help in organizing numbers in a predictable manner. This predictability allows us to examine properties such as patterns, limits, and behaviors of the numbers.

Finite Differences

Finite differences involve calculating the change between consecutive terms in a sequence. It is a simple yet powerful concept used in mathematics to identify and analyze patterns within sequences and series.

For sequences, the finite difference between terms \( f_j \) and \( f_{j-1} \) can be expressed as:

• \( v_j = f_j - f_{j-1} \)

In our example, the formula for the sequence is used to find a formula for the differences:

• The formula for \( v_j \) was derived as \( 2j \).

Evaluating these differences helps understand how quickly the sequence changes as \( j \) increases. Finite differences can reveal linearity or other mathematical properties in sequences.

Formulas

Mathematical formulas are expressions that define the relationships between variables or sequences. In dealing with sequences, formulas help predict the value of terms without having to compute each one individually.

In the given exercise, we started with:

• \( f_j = j^2 + j + 1 \), which expresses the relationship of sequence \( f_j \).

Using this, a new formula for the difference was derived:

• \( v_j = 2j \).

This shows that each time \( j \) increases by 1, \( v_j \) increases by 2, demonstrating a linear relationship for differences.

Formulas are fundamental in understanding sequences as they provide an algebraic method for analyzing and predicting sequences, making them manageable and useful for solving problems.

Subscripts

Subscripts are numbers or variables that appear at the bottom of characters to distinguish similar items in sequences and functions. They act as indexes that indicate the position of terms in a sequence.

In mathematics, especially in sequences, subscripts are critical for identification.

• For example, in \( f_j \), \( j \) serves as the subscript.
• This tells us which term of the sequence we are referring to.

Subscripts allow clarity and precision, making it easier to reference particular elements within a series or mathematical model.

In the context of finite differences, subscripts like \( f_{j-1} \) are used to denote terms before or after a particular point. Without these, keeping track of position in sequences would become complex and error-prone.