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Chapter 3: Problem 6

Look for a pattern and predict the next two numbers in each sequence. $$10,12,16,22,30, \dots$$

Short Answer

Expert verified
The next two numbers in the sequence are 40 and 52

Step by step solution

01

Identify Incremental Changes

Start by looking at the differences between each consecutive pair of numbers. From 10 to 12 it's +2, from 12 to 16 it's +4, from 16 to 22 it's +6, and from 22 to 30 it's +8.
02

Find the Pattern

The differences between the numbers are increasing by 2 each time. This gives us an important insight into the pattern: it seems to be based on growing increments of 2.
03

Apply the Pattern

Continuing this pattern, the increment to the next number in the sequence should be +10 (because 8+2 = 10). So, 30 + 10 = 40, which suggests that 40 is the next number in the sequence. For the following number, we increment by +12 (because 10+2 = 12), so 40 + 12 = 52 which suggests that 52 is the subsequent number in the sequence.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sequence Prediction
When attempting to predict the continuation of a number sequence, the first step is to closely examine the pattern formed by existing numbers. Sequence prediction helps us understand the logic or rule that drives a series and allows us to forecast future elements.
Begin by listing the numbers: 10, 12, 16, 22, 30. Notice how they change. This shift between numbers holds the key to predicting what comes next. The process typically involves:
  • Identifying consistent differences or operations between terms.
  • Looking for repetition or patterns within these operations.
  • Utilizing these patterns to extend the sequence.
For example, in this particular sequence, the differences between numbers progressively increase, setting a predictable pattern for further numbers.
Incremental Changes
Analyzing incremental changes lets us understand how the numbers differ from one step to the next. This involves calculating the change between each consecutive number and identifying any patterns in these changes. For the sequence 10, 12, 16, 22, 30, the changes are:
- From 10 to 12: +2 - From 12 to 16: +4 - From 16 to 22: +6 - From 22 to 30: +8
These increments are not constant but increase steadily. Recognizing this pattern is crucial. The difference itself grows by 2 each time, indicating that the series adds an extra 2 to each subsequent change.
Mathematical Reasoning
Mathematical reasoning involves using logical thinking to understand and extend patterns. It allows us to move from observing a sequence to predicting and verifying subsequent numbers with certainty.
In this sequence, the increment was increasing by 2 each step. Applying this reasoning:
  • The next increment would be +10, thus 30 + 10 = 40.
  • Following this, increase by +12, so 40 + 12 = 52.
Mathematical reasoning not only helps us complete sequences but also deepens our comprehension of mathematical structures and relationships. With these logical steps, you can accurately complete the sequence and reinforce understanding of number patterns.

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Most popular questions from this chapter

Draw several diagrams to help you decide whether each statement is true or false. If it is false, show a counterexample. If it is true, draw and label a diagram you could use in a proof. List, in terms of the diagram, what is given and what is to be proved. Do not write a proof. If a triangle has two congruent angles, then the sides opposite those angles are congruent.

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