Notice that the word is guess. That's the best you can do for inductive reasoning. Whatever pattern you make up, someone else might devise a different pattern.

Here are a few other common strategies for finding the next number in a sequence. Consider the sequence 1, 3, 5, 7, 9, 11, 13, . . . What is the next number in the sequence? You might recognize the sequence as a list of the odd numbers. Then next number would be 15. If you didn't recognize the odd numbers, you might try looking for a pattern. You would see that the numbers go up by 2 each time. So the next number would be 13 + 2 = 15.

Consider this sequence: 1, 4, 9, 16, 25, ... What is the next number in the sequence? You might recognize that this is a sequence of perfect squares.

The first term is 1 x 1 = 1. The second term is 2 x 2 = 4. The third term is 3 x 3 = 9. The fourth term is 4 x 4 = 16. The fifth term is 5 x 5 = 25.

Predict that the pattern would continue, and the sixth term would be 6 x 6 = 36.

Consider this sequence: 1, 3, 6, 10, ... Follow these steps:

1. Identify the question. You need to find the next number in the sequence.

2. Note all of the facts you already have. You have four terms already. They appear to be getting larger. They are not consecutive numbers. They are not the perfect squares.

3. Determine what is missing. You don't know the pattern.

4. Do some creative work. Take a look at how each term is built from the one before it.

You might add 1 + 2 to get to 3, but 3 + 2 is not 6. 3 + 3 is 6. What do you add to 6 to get to 10? Well, it's 4. Write the numbers in a table, and take a look at the pattern.

A | |||

'il | |||

1+ |
2 |
= |
3 |

3+ |
3 |
= |
6 |

6+ |
4 |
= |
10 |

Iy |

Now you can see the pattern. Each time, you add the next counting number to the term. Often a visual aid such as creating a table helps you identify a pattern you may not notice in paragraph form.

To help build spatial-relationship skills, view these numbers in a picture.

The triangular arrangement prompts us to name this sequence the triangular numbers. Looking at the pattern of dots, decide how adding a counting number each time adds dots to the figures.

The next exercise requires that you perform long division. You may prefer to use a calculator, but the display may not hold enough digits for you to recognize the pattern. Complete the following table and then look for patterns. Notice that the division will never end, so stop after 12 digits in your answers. As soon as you recognize the pattern, test one more number, and then fill in the Quotient block from the pattern. Answers are at the end of this chapter.

Division Quotient

Notice the patterns within the quotients that relate to 7? Examine the Quotient column. Do you see that there is a 14 and 28 in each number? That is 2 x 7 and 4 x 7. Another pattern is that the first digits in the quotients are in ascending order. If you remember the sequence of numbers in the repeating section, then you can dazzle your friends and confound your enemies with math magic tricks and lightning-speed division prowess!

Not all sequences involve numbers. Consider this sequence of letters: S, M, T, W, T, F, S

They are the first letters of the days of the week. What do you think the next letter is in the following sequence of letters?

These are the first letters of the words for the counting numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...

Sequence of Letters

Here's another: A, E, F, H, ... What is the next letter in this sequence? _____The answer is at the end of this chapter.

This sequence is another favorite for building pattern-recognition powers. In the blanks in the table, write the counting numbers down the first column and then up the second column. Do this before you go on to the next paragraph.

Take some time to study the list of numbers that you generated. What do you recognize about the list of numbers? Do this before you go on to the next paragraph.

Did you recognize this as the nine times table? The pattern that is generated by the nine times table yields great fun. Try this: Add the digits across each row in the table. For example, for the row containing 18, add 1 + 8. Then 2 + 7, and so on.

You found another pattern. The sum is always 9. Here is the fun part. Because the numbers always add up to 9, you only need nine fingers to represent the numbers in the table. You can use your hands to multiply by 9. Consider the diagram in Figure 7-2. Place your hands on the table in the same manner.

Suppose you want to show 4 x 9. Tuck the finger numbered 4, the index finger of your left hand, under your hand. This may be difficult at first if you have arthritis in your hands, but the exercise may improve your flexibility! There are 3 fingers up on the left of finger #4, and there are 6 to the right. 3, 6: The answer to 4 x 9 is 36. See Figure 7-3.

3 fingers 6 fingers

3 fingers 6 fingers

Try this again. This time you'll multiply 6 x 9. First tuck finger #6, the thumb of your right hand, under your hand, as in Figure 7-4. There are 5 fingers up on the left of finger number 6 and 4 fingers up on the right of finger number 6 Six times nine is 54.

5 fingers

5 fingers

Figure 7-4 6 times 9 is 54

4 fingers

4 fingers

Figure 7-4 6 times 9 is 54

This is something you might share with someone else. Practice a few times by yourself, and then teach this trick to a friend or a child who is learning the multiplication tables.

Inductive Reasoning Exercises

In each problem, write the next term in the sequence. Also, write the reason why you think that is the next term.

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