Find the essence of the law problem: find the corresponding relationship between the numbers in the sequence and their serial numbers.

1, arithmetic type.

Compare each number with its previous number. If the difference is constant (usually called tolerance), then the nth number can be expressed as an=a 1+(n- 1)d, where a 1 is the first number in the series and d is the difference, (n- 1).

Example 1, 3, 6, 9, 12 ... Find the nth number.

Solutions; Starting from the second number, each number is 3 more than the previous number and 3 worse, so the nth digit is: 3+(n- 1)×3=3n.

2. What is added is arithmetic.

In other words, each increase is compared with the previous increase, and the difference between the two increases is unchanged.

3. Equal ratio type.

Compare each number with its previous number. If the ratio is constant, then the nth number can be expressed as an=a 1qn- 1, where a 1 is the first number in the series and q is the ratio.

Examples 5, 3, 6, 12, 24 ... Find the nth number.

Solutions; Starting from the second number, the ratio of each number to the previous number is always 2, so the nth bit is 3×2n- 1.

4. The growth rate is equal.

In other words, every increase is compared with the previous increase, and the increase ratio is unchanged.

Example 6, 2, 3, 5, 9 ... What is the eighth item in this series?

Solution: Starting from the second beam, the increments of each number and the previous number are 1, 2, 4 and 8 respectively, so the sixth number is 17+24=33, the seventh number is 33+25=55, and the eighth number is 55+26 =1.

5. Square type: the sequence is the square of each serial number, the square of serial number+constant, and the square-constant of serial number.

Example 7. We know that the first few terms of a series are 2,5, 10, 17 ... What is the nth term of the series?

Solution: According to the observation, the first items of the sequence are equal to 12+ 1, 22+ 1, 32+ 1, 42+ 1, so it can be inferred that the nth item is n2+ 1.

Example 8. Observe the following numbers: 0, 3, 8, 15, 24 ... Try to write the number 100th according to this rule.

Solution: According to observation, the first term of the series is equal to 12- 1, 22- 1, 32- 1, 42- 1, so it can be inferred that the nth term is n2- 1.

100th is: 1002- 1 = 9999.

6. Index.

Example 9. Observe the following numbers: 1, 2, 4, 8, 16 ... Try to write the number 1 1 according to this rule.

Solution: According to the observation, the first few terms of the sequence are equal to 20, 2 1, 22, 23 ... Then it can be inferred that the nth term is 2n- 1.

1 1 This number is: 2 10? = 1024。