Compare the neighboring terms in the sequence and find the difference between each pair of neighboring terms. List the differences as a sequence and try to identify the nature of the pattern formed by the differences. Use the pattern of differences to formulate a rule that predicts the new term in the sequence. For example, the number pattern 2, 5, 10, 17, 26 yields a sequence of differences 3, 5, 7, 9. The pattern formed by the differences between the terms of the original sequence is a list of increasing odd numbers. To find the next number in the original sequence, find the next number in the sequence of differences and add that difference to the last term in the original sequence. For example, in the sequence 2, 5, 10, 17, 26 the next term will be 37. The term 37 is obtained by adding the next difference (11) in the pattern of odd numbers to the last number in the original pattern (26): 37 = 26 + 11. If no obvious pattern emerges from the study of the sequence of differences between the terms, proceed to generate a different rule that describes the pattern.
Write each term in the pattern as a set of its prime multiples. For example, the number 27 should be written as 27 = 3 x 3 x 3, while the number 21 would be written as 21 = 3 x 7.
Create a table with one more column than the largest number of prime multiples found for any one of the terms in the pattern. Use the first column to list all the terms. Use subsequent columns to list the prime multiples of each term. Each term and its multiples are arranged in rows in the table. Arrange the multiples beneath each other so that like multiples are aligned vertically in columns. Fill all the open cells with the multiple 1. For example, the pattern of 3, 9, 18, 30, 45, 63, 84 would require a table that has 7 rows (for 7 terms) and 7 columns. The terms in this pattern are expanded to the following sets of multiples: 3 = 3 x 1 x 1 x 1 x 1 x 1; 9 = 3 x 1 x 3 x 1 x 1 x 1; 18 = 3 x 2 x 3 x1 x 1 x 1; 30 = 3 x 2 x 1 x 5 x 1 x 1; 45 = 3 x 1 x 3 x 5 x 1 x 1; 63 = 3 x 1 x 3 x 1 x 7 x1; 84 = 3 x 2 x 1 x 1 x 7 x 2.
Study the table to see if the arrangement of multiples forms a recognizable pattern. Use the pattern to formulate a rule. For example, the terms in the pattern of 3, 9, 18, 30, 45, 63, 84 all share a common multiple of 3. If the highest prime multiple for each term is isolated, it forms the pattern 1, 3, 3, 5, 5, 7, 7. If the remaining multiples for each term are grouped, the sequence formed is: 1, 1, 2, 2, 3, 3, 4. The term following 84 in the original pattern is predicted to be 108. This is obtained by finding the next term in each of the sub-sequences identified by inspection of the table and multiplying those terms as you would multiples. For example, the next term in the odd-number sequence of 1, 3, 3, 5, 5, 7, 7 would be 9. The next term in the sequence of 1, 1, 2, 2, 3, 3, 4 would be another 4, since all the other terms appear in pairs. Finding the next number in the original pattern is then a process of applying the observed rule, which involves multiplying 3 (the multiple common to all the terms), 9 and 4. This yields 3 x 9 x 4 = 108.