1. Fill in the blanks (for each small question 10, ***60)

1. Group natural numbers in order: the first group contains one number, the second group contains two numbers, the third group contains three numbers, ..., and the nth group contains n numbers, that is,1; 2,3; 4,5,6; ..... Let an be the sum of the nth group, then an = _ _ _ _ _ _ _ _ _ _ _ _

2.=______________.

3.=_________________.

4. It is known that the bottom surface of a parallelepiped is a rhombus, and its acute angle is equal to 60 degrees. The side of this acute angle forms an equal angle with both sides of the acute angle and forms an angle of 60 degrees with the bottom surface, so the ratio of the areas of the two diagonal surfaces is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

5. Positive real numbers x and y satisfy the relation x2? xy？ 4 = 0, and if x≤ 1, the minimum value of y is _ _ _ _ _ _ _.

6. A 500-meter-long train runs at a constant speed on a straight track. When the tail of the train passes a platform, someone drives a motorcycle to chase the train out of the platform and send an emergency message to the train driver, and then return at the original speed. On the way back, when he met the rear of the train, he found it was 0/000 meters away from the platform/kloc-. Assuming that the speed of the motorcycle is constant, the motorcycle has traveled _ _ _ _ _ _ _ _ from the departure to the platform.

Second, solve the problem (each small question 15, ***90)

The 1. sequence {an} is applicable to the recurrence formulas An+ 1 = 3an+4, A 1 = 1. Find the first n terms and Sn of the sequence.

2. It is proved that the light emitted from the elliptical focal point must pass through another focal point after being reflected by the smooth elliptical wall. Do you know the optical properties of other conic curves? Please describe it but don't have to prove it.

3. The height of the regular hexagonal cone is greater than h, and the included angle between two adjacent sides is equal to,

Find the volume of the pyramid. ()

4. Let z 1, Z2, Z3 and Z4 be four points on the unit circle on the complex plane. If z 1+Z2+Z3+Z4 = 0.

Proof: These four points form a rectangle.

5. Let, where xn and yn are integers, find the limit of n→∞.

6. Suppose there are three points on a plane, and the distance between any two points does not exceed 1. Q: What is the minimum radius of a disk covering these three points? Please prove your conclusion.