first step

Overall observation, if there is a linear trend, take the idea A, if there is no linear trend or the linear trend is not obvious, take the idea B.

Note: the linear trend is that the exponential series generally develops in one direction, that is, the numerical value is getting bigger and bigger, or getting smaller and smaller, and the change of numerical value is directly related to the number of items (don't feel too mysterious, in fact, everyone can have this intuition after doing some questions).

Second step

According to the analysis in the first step, choose idea A or B.

Idea A: Analyze the trend

1, increase (including decrease) is generally addition and subtraction.

The basic method is to do it almost, but if you can't find the rule by doing it three levels worse, you should change your mind immediately, because arithmetic progression and his variants above level three failed the public examination.

Examples 1:-8, 15, 39, 65, 94, 128, 170, ()

180

Solution: the observation is linear, and the value increases gradually, and the increase is general. Considering the difference, you have to take 23, 24, 26, 29, 34, 42 to form a linear sequence of small increments again, and then do the difference to get 1, 2, 3, 5, 8, an obvious and recursive sequence, and the next item is 5+8.

Summary: the difference will not exceed level 3; Some typical series should be memorized.

2, do the multiplication and division method with large increase.

Example 2: 0.25, 0.25, 0.5, 2 16, ()

64 C. 128 D.256

Solution: the observation shows a linear law, increasing from 0.25 to 16, considering multiplication and division. The latter item is divided by the former item to get 1, 2, 4, 8. Typical geometric series, the next term of the second-order series is 8*2= 16, so the next term of the original series is 16 * 655.

Summary: Doing business will not exceed level 3.

3, the increase is very large. Consider power series.

Example 3: 2, 5, 28, 257, ()

2006. 1342 C .3503 D .3 126

Solution: the observation is linear and the increase is very large. Considering power series, the law of maximum number is obviously the breakthrough of this problem. Note that there are powers of 256 near 257, 4 and 8 near 27, 25 and 28, and 1 and 4 near 2. And each term of a series must be related to the number of terms, so the power series related to the original series should be 1, 4,27,256 (each term of the original series is added with 1, that is,1,22,3,4), and the next term should be 5.

Summary: Familiar with power numbers.

Idea B: Find the visual impact point.

Note: Visual impact point is a special and unusual phenomenon in exponential series, which is often the direction of solving problems.

Visual impact point 1: a long sequence of more than 6 items. The basic idea to solve the problem is to group or separate the projects.

Example 4: 1, 2, 7, 13, 49, 24, 343, ()

A.35 B .69 degrees Celsius. 1 14 D .238

Solution: The first six items are relatively small, and the seventh item suddenly becomes larger, which is not a linear law. Consider idea B. For long series, consider grouping or separation, and try two separation series: 1, 7, 49, 343; 2, 13,24,()。 Obviously, the first branch series is a geometric series, and the second branch series is a arithmetic progression with a tolerance of 1 1, and the answer A is quickly obtained.

Conclusion: Interval mixing of arithmetic and geometric progression is a common test.

Visual impact point 2: swing series, numerical fluctuation, showing a swing shape. The basic idea of solving problems is sub-topic.

20 5

Example 5: 64, 24, 44, 34, 39, ()

10

A.20 B .32 degrees Celsius 36.5 degrees. 19

Solution: the observed values are small and large, and every other item is observed immediately. If the difference is as above, it is found that the difference becomes a geometric series, and the next difference should be 5/2=2.5, and the easy answer is 36.5.

Summary: The number of items is not necessarily regular, but it is also possible to form a comprehensive law like this problem.

Visual impact point 3: double brackets. There must be a regular pattern!

Example 6: 1, 3, 3, 5, 7, 9, 13, 15, (), ()

A.19,21b.19,23 degrees celsius.21.23d.27,30

Solution: Find the rules directly when you see the double brackets, and use 1, 3,7, 13, (); 3, 5, 9, 15, (), which is obviously a two-level arithmetic progression with a tolerance of 2, easy to answer 2 1 23, and choose C.

Visual impact point 4: score.

Type (1): Mix and match integers and fractions, prompting multiplication and division.

Example 7: 1200, 200, 40, (), 10/3

10 .20 degrees Celsius. Thirty days. Five

Solution: Mix and match integers and fractions, and immediately associate them with quotient, so it is easy to get the answer of 10.

Visual impact point 5: positive and negative overlap. The basic idea is doing business.

Example 8: 8/9, -2/3, 1/2, -3/8, ()

A 9/32 B 5/72 C 8/32 D 9/23

Solution: positive and negative overlap, do business immediately, and find that it is a geometric series, and it is easy to get an A.

Visual impact point 6: radical.

Roots and integers are mixed in the type (1) series. The basic idea is to turn integers into roots and move numbers outside the roots into roots.

Example 9: 0316 √ 212 () () 2 48

A.√3 24 b√3 36 c . 2 24d . 2 36

Solution: There are 0, 1, √2, (), 2 before the double brackets; 3, 6, 12, (), 48. The first branch number is a mixture of a root number and an integer, with √2 as the base number, and other numbers are deformed around it, and the integer is √0 √ 1 √2 ()√4, which is easy to know. The second branch series is a geometric series with obvious common ratio of 2, so the answer is a.

Visual impact point 8: Pure decimal series, that is, all items in the series are decimals. The basic idea is to consider the integer part and the decimal part separately, or form a separate series or * * * formula with the same law.

Example10:1.01.1.02, 2.03, 3.05, 5.08, ()

A.8. 13 B .8.0 13 degrees celsius.7.12

Solution: The integer part is extracted by 1, 1, 2, 3, 5, (), which is an obvious and recursive sequence, and the next item is 8, excluding C and D; Extract the decimal part, including 1, 2, 3, 5, 8, () and recursive sequence, and the next item is 13, so choose A.

Summary: This question belongs to the independent law of integer and decimal parts.

Visual impact point 9: natural number, there are more than three natural numbers in the sequence. Because the operation intensity of the sequence problem is not strong, it is impossible to use natural numbers to do operations, so this kind of problems generally investigate the microscopic number structure.

Example 18: 76395 1, 59367, 7695, 967, ()

69 degrees Celsius in 5936 AD. 769.76 AD.

Solution: It is not practical to calculate natural numbers. Microscopically, it is found that the last item is one digit less than the last item, at least 1, 3,5, and the next default number should be 7; In addition, after one digit is defaulted, the numerical order is also reversed, so it is reversed after 967 is divided by 7. It should be 69, and I chose B.

Third step

Find another way.

Generally speaking, it is the last two steps, and most of the questions can find ideas, but it does not rule out that some laws are not easy to find directly. At this time, if the original series is slightly changed, it may be easier to see the law.

Variant 1: common factor decreases. The number of series is large and there is a common divisor. You can remove the common divisor first, transform it into a new series, and then recover it after finding the law.

Example 20: 0, 6, 24, 60, 120, ()

A. 186 .2 10℃ for 220 days.226

Solution: Because each value of this series is very large, we are not sure whether the increase is large or small, but we find that there is a common divisor of 6, which is rounded to get 0, 1, 4, 10, 20. It is easy to find that the increase is average. Considering addition and subtraction, we can easily find that it is a second-order arithmetic progression, and the next term should be 20+ 10+.

Variant 2: Factorization. Each item in the sequence does not have the same divisor, but adjacent items have the same divisor. At this time, factorizing the numbers in the original series is helpful to find the law.

Example 2 1: 2, 12, 36, 80, ()

100 . 125 C 150 D . 175

Solution: Factorization includes 1*2, 2*2*3, 2 * 2 * 5, and it is easy to get 1* 1*2, 2*2*3, 3 * with a little modification.

Variant 3: General scoring method. There are not many least common multiples of the denominator applicable to each item in the fractional column.

Example 22: 1/6, 2/3, 3/2, 8/3, ()

10/3 B.25/6 C.5 D.35/6

Solution: It is found that the denominator is easy to divide, and the single molecular sequence 1, 4,9, 16, () can be obtained by dividing the denominator immediately. The increase is general, with a difference of 3, 5 and 7, and the next item should be 16+9=25. The score with reducing component 6 is b.