S odd /S even = (n+ 1)/n?

Let the first term of the original sequence be a and the tolerance be d,

The original order is a, a+d, a+2d, a+3d. ，a+2nd。

Odd terms are: a, a+2d, a+4d,. ，a+2nd。

Sum of odd terms: s odd number = [a+(a+2nd)] (n+1)/2 = (a+nd) (n+1)

Even terms are: a+d, a+3d, a+5d,., a+(2n-1) d.

Sum of even terms: even = [(a+d)+(a+2nd-d)] n/2 = (a+nd) n.

S odd /S even = (n+ 1)/n?

Arithmetic progression refers to the sequence of the same constant whose difference between each term and the previous term is equal to the second term, usually expressed by a and p ... This constant is called arithmetic progression's tolerance, often expressed by the letter D.

For example: 1, 3, 5, 7, 9...2n- 1. The general formula is: an = a1+(n-1) * D. The first term a 1= 1, and the tolerance d=2. The first n terms and formulas are: sn = a1* n+[n * (n-1) * d]/2 or Sn=[n*(a 1+an)]/2. Note: All the above n are positive integers.

(1) As can be seen from the general formula, a(n) is a linear function or a constant function (d=0) of n (d≠0), and (n, an) is arranged in a straight line. According to the previous n terms and formulas, S(n) is a quadratic function of n (d≠0

(2) From the definition and general formula of arithmetic progression, we can also deduce the first n terms and formulas: a (1)+a (n) = a (2)+a (n-1) = a (3)+a (n-2) = … = a (k). . . =p(k)+p(n-k+ 1))，k∈{ 1，2，…，n} .

(3) If m, n, p, q∈N* and m+n=p+q, then a(m)+a(n)=a(p)+a(q), and s (2n- 1) = (2n-65438). If m+n=2p, then a(m)+a(n)=2*a(p).

Prove: p (m)+p (n) = b (0)+b (1) * m+b (01) * n = 2 * b (0)+b (1) * (m

p(p)+p(q)= b(0)+b( 1)* p+b(0)+b( 1)* q = 2 * b(0)+b( 1)*(p+q)； Because m+n=p+q, p (m)+p (m)+p (n) = p (p)+p)+p.

(4) Other inferences:

① Sum = (the first item+the last item) × the number of items ÷ 2;

② Number of items = (last item-first item) ÷ tolerance+1;

③ The first term =2x and the number of terms-the last term or the last term-tolerance× (the number of terms-1);

④ The last item =2x and the number of items-the first item;

(5) the last item = the first item+(item number-1)× tolerance;

⑥2 (sum of the first 2n terms and-the first n terms) = sum of the first n terms and+the first 3n terms and-the first 2n terms.

According to historical legends, chess originated in ancient India, and the earliest record seen in the current literature was written in Persian during the Sassanian Dynasty. It is said that a Hindu prime minister saw the king's conceit and vanity and decided to teach him a lesson. He recommended a game to the king that was unknown at that time. At that time, the king was surrounded by a group of ministers who flattered him all day. He was bored and needed games to relieve his depressed mood.

The king soon became interested in this novel game. When he was happy, he asked the Prime Minister what reward he needed as a reward for his loyalty.

The Prime Minister said: Please put 1 grain of wheat in the first, second, third and fourth squares on the chessboard ... that is, the number of grains put in the back grid must be twice as much as that in the front grid at a time, until the 64th grid in the last grid is full, so I am very satisfied. "All right!" The king smiled and generously agreed to the humble request of the Prime Minister.

How many grains of wheat did the clever Prime Minister ask for? After a little calculation, we can get:1+2+22+23+24+...+263 = 264-1,and the number is directly18,446,744,073,709.

If a granary with a width of four meters and a height of four meters is built to store these grains, the granary will be 300 million kilometers long, which can go around the equator of the earth 7500 times, or go back and forth between the sun and the earth once.

Where does the king have so much wheat? His generous words became a debt that he could never repay to Prime Minister Sass Bandal.

Just when the king was at a loss, the prince's math teacher learned about it. He smiled and said to the king, "Your Majesty, this question is very simple, as easy as 1+ 1=2. How can you be stumped by it? "

The king was furious: "Do you want me to give him all the wheat produced in the world in 2000?" The young teacher said, "There is no need, Your Majesty. In fact, all you have to do is let the Prime Minister go to the granary and count the wheat himself. If the Prime Minister counts one grain per second, it will take about 580 billion years to finish counting18,446,744,073,709,551,6 15 grains of wheat (you can do it yourself with a calculator! )。

Even though the Prime Minister counted it day and night until he entered the Elysium, he only counted a tiny part of those grains. In this case, it is not that your majesty can't pay the reward, but that the prime minister himself can't take it away. "The king suddenly realized that he now called the prime minister and told him the teacher's method.

Sass Bandal thought for a moment and then smiled: "Your Majesty, your wisdom has surpassed mine. Those rewards … I have to give up!" " "Of course, in the end, the Prime Minister got a lot of rewards.

The arithmetic mean term is half of the sum of arithmetic progression's head and tail terms, but you don't have to know the head and tail terms to find the arithmetic mean term. In arithmetic progression, the arithmetic mean is generally set to A(r). When A(m), A(r) and A(n) become arithmetic progression, A(m)+A(n)=2×A(r), so A(r) is the arithmetic average of A(m) and A(n) and the average of the series. And it can be inferred that n+m=2×r, and the relationship between any two terms a(m) and a(n) is: a(n)=a(m)+(n-m)*d, (similar to p (n) = p (m)+(n-m) * b (6544).

Application of arithmetic progression In daily life, people often use arithmetic progression. For example, when grading the sizes of various products, when the maximum size is not much different from the minimum size, arithmetic progression is often used for grading. If it is arithmetic progression, and a (n) = m and a (m) = n, then a(m+n)=0.

In fact, Zhang Qiujian of China in the ancient Southern and Northern Dynasties has already mentioned arithmetic progression in Zhang Qiujian's mathematical masterpieces: Today, some women are not good at weaving, and the amount of cloth they weave every day has decreased by the same amount. At the beginning of the day, they knit five feet, and at the end of the day, they knit for thirty days. Ask * * * to weave geometry? The solution in the book is: combine the first and last knitting numbers, half, and the rest according to the number of knitting days. This is equivalent to giving the summation formula of s (n) = (a (1)+a (n))/2 * n.