Problem 48 - 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. $$ \begin{array}{r} 3+6=\frac{6 \times 3}{2} \\ 3+6+9=\frac{9 \times 4}{2} \\ 3+6+9+12=\frac{12 \times 5}{2} \\ 3+6+9+12+15=\frac{15 \times 6}{2} \end{array} $$

Step-by-Step Solution

Answer

From the observed pattern, the next line of the sequence is: $3+6+9+12+15+18=\frac{18*7}{2}$. After performing the arithmetic calculations, we've verified that both sides of the equation hold true.

Step 1: Identify the patterns

Firstly identify the patterns used in the given sequences. On the left side, the numbers are incremented by adding 3 to each number: 3, 6, 9, 12, 15. On the right side, the highest number from the left side of the equation is used, then multiplied by an increasing positive integer and divided by 2: \(\frac{3*2}{2}\), \(\frac{6*3}{2}\), \(\frac{9*4}{2}\), \(\frac{12*5}{2}\), \(\frac{15*6}{2}\)

Step 2: Apply the pattern to find the next line

From step 1, the next logical number in the left-side sequence is 18 (15+3). On the right side, the next number that the highest number should be multiplied by is 7 (increasing 1 from the last number 6), and then divided by 2, which gives us \(\frac{18*7}{2}\)

Step 3: Check with Calculation

Now, do the calculation for the right-side equation: \(\frac{18*7}{2}=63\). Also, compute the addition for the left-side sequence: 3+6+9+12+15+18=63. Both sides are equal, which validates our pattern detection and application.

Key Concepts

patterns in sequences

Patterns in sequences are essentially the repeating elements that appear consistently in a specific order. These patterns are crucial for solving problems that involve predicting future elements or making mathematical conjectures. In the exercise provided, we observe a pattern in both the sequence of numbers on the left side of the equation and the arithmetic operations on the right.

• On the left, the numbers increase by 3 each time: 3, 6, 9, 12, 15.
• Meanwhile, the numbers on the right follow a formula involving the largest number from the left sequence.

This detailed pattern recognition becomes a tool for prediction. By determining the consistent difference or operational steps, you can predict the next numbers in the sequence. Recognizing patterns is key to solving many mathematical problems efficiently. It involves a keen observation of any repeating sequences or operations that allow prediction and easy identification of future elements.

arithmetic progression

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This is one of the simplest types of numeric sequences and is commonly encountered in everyday math problems. In the given exercise, the sequence on the left follows an arithmetic progression.
An arithmetic progression can be represented algebraically as:
- First term (\(a_1\)) - Common difference (\(d\)) - General term (\(a_n = a_1 + (n-1)d\))
In our exercise:

• The first term is 3.
• The common difference is also 3.
• This results in the sequence: 3, 6, 9, 12, 15...

Arithmetic progressions are foundational in algebra and help form further complex mathematical theories. Understanding this progression aids in solving for unknowns in sequences and ensuring predictions are consistent with discovered patterns.

mathematical conjecture

A mathematical conjecture is a proposition or conclusion based on incomplete information, for which no definitive proof has been found. It often serves as a starting point for further investigation and proof. In the exercise, inductive reasoning is used to derive a conjecture about the sequence of computations. This means using observed patterns to generalize and predict the next elements in the sequence.
To form a conjecture, one can follow these steps:

• Observe a pattern or a series of points.
• Formulate a hypothesis predicting the continuation of this series.
• Test the hypothesis using new data or further computation.

In this scenario, the process involved recognizing a consistent relationship between the sums and the formula on the right side, predicting the next equation, and verifying it with computation. This iterative process allows for the exploration and proof of mathematical truths through careful analysis and application of noted patterns.