Problem 50 - 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 \(1 \times 9+(1+9)=19\) \(2 \times 9+(2+9)=29\) \(3 \times 9+(3+9)=39\) \(4 \times 9+(4+9)=49\)

Step-by-Step Solution

Answer

The next line in the sequence is \(5 \times 9+(5+9)=59\).

Step 1: Comprehend the Pattern and Create a Formula

Understand the sequence and you will notice a pattern. Each operation seems to follow the formula \(x \times 9+(x+9)\) where \(x\) is the integer present on the left most side of the operation.

Step 2: Conjecture the Next Line

Based on the established pattern, predict the next line. With \(x = 5\) (as the last number was 4 in the sequence), the next line in the sequence therefore will be \(5 \times 9+(5+9)=59\). Use this conjecture to compare with the calculated result.

Step 3: Verify Conjecture

Determine whether the conjecture is correct by performing the arithmetic. Calculate \(5 \times 9+(5+9)\) and compare this with the predicted value. After calculations, you will find the result equals \(59\). This confirms the conjecture.

Key Concepts

Pattern Recognition

Recognizing patterns is an essential skill in mathematics, and it plays a key role in understanding sequences of computations. In the given exercise, the first step requires identifying the formula used in the sequence. The sequence is:

• \(1 \times 9 + (1+9) = 19\)
• \(2 \times 9 + (2+9) = 29\)
• \(3 \times 9 + (3+9) = 39\)
• \(4 \times 9 + (4+9) = 49\)

Notice how each line follows a similar structure. This pattern can be distilled into a general formula: \(x \times 9 + (x+9)\), where \(x\) begins at 1 and increases by 1 each time. This repeated pattern across different operations is key to predicting subsequent results. By seeing how these numbers interact, students can identify the underlying formula and comprehend what each component of the expression is doing."

Sequence Prediction

Once a pattern is recognized, the next step is to use this pattern to predict future elements in the sequence. After identifying the pattern or formula used in the problem, you can conjecture what the next line will be. In our example, since we already know the sequence follows \(x \times 9 + (x+9)\), and the last number in the sequence provided is 4, we substitute \(x = 5\) into the formula.If we replace \(x\) with 5, the expression becomes:\[5 \times 9 + (5+9) = 45 + 14 = 59\]This prediction shows how the pattern continues. This step demonstrates the concept of sequence prediction—using a recognized pattern to guess the terms that come next in a series. It's an essential part of inductive reasoning in mathematics, establishing a connection between different elements of a sequence.

Arithmetic Verification

Predicting the next line in the sequence is only the first half of the problem-solving process. To complete it, you must verify your prediction by performing the calculations. This step requires checking if the computed result matches your conjecture. Here, we took our predicted statement:\[5 \times 9 + (5+9) = 59\]Perform the multiplication and addition step-by-step:

• First calculate \(5 \times 9 = 45\)
• Then calculate \(5 + 9 = 14\)
• Finally add these results: \(45 + 14 = 59\)

The resulting value, 59, matches the earlier calculated value. Arithmetic verification displays how important it is to ensure that your predictions are accurate. This is where your understanding of mathematical operations really shines, confirming that the pattern you've recognized correctly describes the sequence.