Sum (1 + a) + (2 + A ^ 2) + (3 + A ^ 3) + +(n + A ^ n) to explain the answer, it's urgent!

=(1+2+3..+n)+a+a^2+...a^n
=n*(n+1)/2+a(1-a^n)/(1-a)

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1. Reading material, mathematician Gauss once studied such a problem in school, 1 + 2 + 3 + +10=? After research, the general conclusion of this problem is 1 + 2 + 3 + +N = n (n + 1), where n is a positive integer. Now let's study a similar problem: 1 × 2 + 2 × 3 + + n（n+1）=? Look at three special equations: 1×2=（1×2×3-0×1×2） 2×3=（2×3×4-1×2×3） 3×4=（3×4×5-2×3×4）. After reading this passage, please calculate: （1）1×2+2×3+… +100 × 101; (just write the result) （2）1×2+2×3+… +N (n + 1); (write the calculation process)
2. Sum an = 1 / 3N (n + 1), Sn
3. Try to judge: are triangles with 2n & # 178; + 2n, 2n + 1,2n & # 178; + 2n + 1 sides right triangles? N is greater than 0
4. Summation of the fifth series is a compulsory course in mathematics of grade one in senior high school The general term formula of sequence {an} is an = (- 1) ^ (n-1) * (4n-3), and the sum of its first n terms can be obtained It must be adopted in time. Please use the multiply common ratio dislocation subtraction method
5. The triangle whose three sides are 2n2 + 2n, 2n + 1, 2n2 + 2n + 1 (n is a positive integer) is () A. Acute triangle B. right triangle C. pure triangle D. acute or right triangle
6. There are two points a and B on the flat ground. A is to the east of mountain D, B is to the southeast of mountain D, and 300 meters in the direction of 25 ° south by west of A. the elevation of C on the top of mountain a is 30 degrees, so as to calculate the mountain height
7. Is a triangle whose three sides are square of a = m - square of N, B = 2Mn, C = square of M + square of n (M greater than n greater than o) a right triangle
8. Calculate 1 + (1 + 2) + (1 + 2 + 3) + +（1+2+3+… +n）．
9. Prove: square of M - square of N, 2Mn, square of M + square of N, (M and N are natural numbers, and M is greater than N and greater than 0) is the side length of right triangle
10. Calculate 1 + (1 + 2) + (1 + 2 + 3) + +（1+2+3+… +n）．
11. 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10.97 + 98 + 99 + 100 =? Gauss's solution
12. 19 + 199 + 1999 + 19999 + 1991 + 2 + 3 + 4 + +97+98+99+100．
13. Sum: (2-3x5 ^ - 1) + (4-3x5 ^ - 2) + +(2n-3x5 ^ - n) urgent!
14. How does Gauss calculate "1 + 2 + 3 +. + 97 + 98 + 99 + 100 = 5050"
15. Let n be a positive integer, can (2n + 1) & sup2; - 1 be divisible by 8? Why?
16. The story of the mathematician Gauss Don't be too short or too long. It'd better be about 70 words. I'm going to do a math paper,
17. It is proved that 8 ^ 2n + 1 + 7 ^ (n + 2) is a multiple of 57
18. The story of mathematician Gauss
19. Can (n + 7) 2 - (N-5) 2 be divisible by 24 when n is a natural number? Give reasons
20. Students, have you heard the story of the mathematician Gauss when he was a child? Do you know how he calculated 1 + 2 = ·· + 99 + 100 = 5050 at once It's due tomorrow