Let me show you one in my way:
1*2+2*3+....99* 100
= sum[n(n+ 1)]= sum[N2+n]= sum(N2)+sum(n)= n(n+ 1)(2n+ 1)/6+n(n+65438)
、
1+7+13+19 ... (* * *100 projects)
= sum (6n-5) = sum (6n)-sum (5) = 6sum (n)+sum (5) = 6 [n (n+1)/2]-5n (n =100 replacement).
4*3+5*3^2+6*3^3+.....(n+3)*3^n
=sum[(n+3)(3^n)]=sum[n 3^n]+3sum(3^n)=sum[(2n- 1)3^(n+ 1)+3]/4+9/2(3^n- 1)
\ The above two examples, I got the results in one breath, and the two questions were wasted only when typing, so I didn't need to calculate them at all. I wrote them down directly and got the results without any thinking.
This is a scientific method of summation of series. Of course, the whole method was created by myself. I can't say clearly here. If you want to learn, you can come to me.
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The following is my own book (unpublished). I was a victim of traditional teaching, and my high school teachers at that time were all parallel imports. Now that I am a teacher, I want to break the traditional teaching routine and ask students not to leave me. The following is my own book, which is an argumentative paper I made while teaching a new method of summation of series. Copy it here, you can have a look.
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Quote my own book:
Traditional methods, such as split term method, grouping method and union method, are actually broken and scattered forms of linear laws, which are flashy and cumbersome, and are swallowed up by summation operators. In addition, the operator's words are concise and the deformation makes people's thinking clear and intuitive.
With PSP and all-encompassing simulator software, why are you still obsessed with FC, MD, SFC, GBA, PS 1? Correct learning attitude and accept it frankly. (flying bricks, flash! )
I often remember that in high school, the Chinese teacher said that a good article should have an echo from beginning to end. In order to completely reverse the views of some biased students, at the end of this chapter, we emphasize again: think about those grouping summation methods, joint summation methods, split summation methods, partial summation methods, reverse summation methods ... There are so many series summation methods with "gorgeous rhetoric", which one is universal? It can only solve the sum of a series.
For example, the reverse summation method, I can't imagine what series it can find besides arithmetic progression. Even if it can ask, who would have thought of it with the thinking mode of human brain calling memory and the environment of examination?
So the traditional messy methods are actually P! These P's can be completely replaced by summation operators with unity/universality/universality. The summation operator method is the "unification" of the summation idea of sequence.
In the process of learning, students should correct their learning attitude, learn to accept advanced things and discard backward things. Never say, "I haven't seen anything, and I certainly don't need to learn." Only what I have seen before is what I want to master.
Finally, we use the summation example of "1* 3+2 * 4+3 * 5+… 99 *1kloc-0/"to completely "end" those old things and let those things that don't keep pace with the times disappear with the rotation of the wheel of history … Thank you! (Bang! I took two steps forward and forgot! )