Problem 47 - Thinking Mathematically

Question

In Exercises 47-52, use inductive reasoning to predict the next line in each sequence of computations. Then use a calculator or perform the arithmetic by hand to determine whether your conjecture is correct. $$ \begin{array}{r} 1+2=\frac{2 \times 3}{2} \\ 1+2+3=\frac{3 \times 4}{2} \\ 1+2+3+4=\frac{4 \times 5}{2} \\ 1+2+3+4+5=\frac{5 \times 6}{2} \end{array} $$

Step-by-Step Solution

Answer

The next line in the sequence is \(1 + 2 + 3 + 4 + 5 + 6 = \frac{6 \times 7}{2}\). The computations validate this prediction, supporting the conjecture that the pattern continues in this manner.

Step 1: Identify the pattern

Identify the pattern followed by the sequence. For every new line, an integer is added to the sequence on the left side of the equation. This new term is equal to the line number. On the right side of the equation, the fraction is calculated by multiplying the line number by one more than the line number, then dividing by 2.

Step 2: Predict the next line using the observed pattern

Using the observed pattern, predict the next line of the sequence: on the left side add the integer 6 to the sequence of integers. On the right side of the equation, calculate the expression by multiplying 6 by 7, and divide by 2.

Step 3: Perform the arithmetic to confirm the prediction

First, compute the sum of the integers on the left side of the equation: \(1 + 2 + 3 + 4 + 5 + 6 = 21.\) Then, perform the calculation on the right side: \(\frac{6 \times 7}{2} = 21.\) The two calculations match, validating the prediction.

Key Concepts

Patterns in Sequences

Patterns are sequences that follow a specific rule or function. Notice, each of our examples involves a set of numbers with some sort of consistent behavior. Recognizing the pattern is critical in using inductive reasoning.

For example, in the sequence given in the exercise:

• Each line introduces a new number to the sum on the left, from 1 to 6.
• On the right, each line takes that new number, adds 1, multiplies them, and divides by 2.

By observing these components, it becomes easier to predict how the sequence continues. Spotting these patterns is an essential skill in mathematics, allowing you to project future data points based on trends.

Arithmetic Sequences

An arithmetic sequence is a collection of numbers where each term is derived by adding a constant value to the previous term. In our case, the sequence on the left grows by adding the next natural number to the sum. Each resulting number is linked through an addition of a constant integer, specifically the next whole number.

Studying these sequences brings to light arithmetic fundamentals:

• Sum Series: The left-hand sequence can be described using sum notation: \(1 + 2 + 3 + \ldots + n\).
• Right-Hand Formula: Using the line number \(n\), the right side expresses the formula as \(\frac{n \times (n+1)}{2}\).

Recognizing arithmetic rules aids in predicting the sequence's next terms and understanding how calculations overlap with these rules.

Mathematical Conjectures

In mathematics, a conjecture serves as an educated guess on the pattern's continuation based on existing observations. It is like a hypothesis; validated only when tested or proven by mathematical logic or calculations. In our problem, conjecturing that the next line conforms to the previously observed pattern means assuming the structure holds true.

To solidify this concept:

• Establish a conjecture based on older examples, e.g., adding the next integer and applying the same right-side formula \(\frac{n \times (n+1)}{2}\).
• Check implementation via calculation to confirm predictions.

Formulating conjectures pushes our understanding further, enabling a logical leap from known data to unknown outcomes, which is a cornerstone in mathematical exploration.