1. This is a very common problem, which will appear in geometric series. There are many varieties. You should try to understand it thoroughly!
Solution: First, write S6 and S3 with equal ratio summation formula, and divide them to get the result.
1-(q to the 6th power) =3-3*(q to the 3rd power)
With method of substitution, let t = (the third power of q), and the above formula will become the well-known quadratic equation of one variable. Then, by shifting terms and cross multiplication, we can get t= 1 or 2. And because (1-q to the third power) undertakes the work of the denominator in the previous part, t= 1 does not hold, so t=2. Then, the following equations S9 and S6 are simplified as described above, and a result can be obtained. I didn't calculate the whole process, so I wrote the steps You have to do the math yourself. But basically this topic will come out. ...
2。 Because it is a geometric series, this topic is very easy to handle, and it is also a classic topic! A3=a 1*q square, a4=a2*q square, so the second formula = the first formula *q square, and the third formula = the second formula *q square. Grab it!
3. Because a 1, a3 and a9 are in equal proportion, (the square of a3) =a 1*a9. And because the three terms are in arithmetic, A3 and A9 can be written as the sum of A 1 and d, and brought back to the above formula for calculation. The result is a 1=d. Then all the items in the title are written in the form of a 1 and D. Because a1= D, it doesn't matter which one it is. Because the D mentioned in the stem is not equal to 0, you can safely and boldly divide the male a 1 or D, and the remaining number is the answer.
There are sufficient conditions for this topic. I think we can write all four items in the most stupid way, and then make a hypothesis one by one. Anyway, there was only one miscalculation, so we calculated them one by one ... The result came out ... I calculated the S3 error.
5.S 10-S9=a 10 (this is a general rule and will be used in general series of questions), so according to the stem of the questions, a 10=a 1 * A9, A9 = A 1 * A8. (this is also a very classic question type, so you should master it. )
6. This is also a classic topic, but it is not very common. Because there are four roots, there should be △ > 0, and it is four different solutions! Then ... I forgot ... I'm ashamed of heaven!
This is probably the key point of this problem ... it seems that there is another one to make sure that the four schemes are really different, and this one seems to be more critical. After all, it is equal proportion. You can try to solve the problem yourself from the characteristics or principles of geometric series. ...
These questions are classic questions in the series, so we must understand them thoroughly! It will be good for the future! !