If cN0 + 1 / 2cn1 + 1 / 3cn2 +... + 1 / (n + 1) CNN = 31 / (n + 1), find the term with the largest coefficient in the expansion of (1-2x) 2n

First, we prove that C (K + 1, N + 1) / C (k, n) = [(n + 1)! / (K + 1)! * (n-k)!] / [n! / K! * (n-k)!] = (n + 1) / (K + 1) so 1 / (K + 1) C (k, n) = 1 / (n + 1) C (K + 1, N + 1) so cN0 + 1 / 2cn1 + 1 / 3cn2 +... + 1 / (n + 1) CNN = 1 / (n + 1) * [C (1

Cn0-2cn1 + 3cn2 +... + (- 1) ^ n (n + 1) CNN =? 1
Is it divided into parity discussion

See this type of problem, the first reaction is to use the binomial theorem
Write the original form as
C(n,0)-2xC(n,1)+3x^2C(n,2)-...
=x'C(n,0)-(x^2)'C(n,1)+(x^3)'C(n,2)-...
=[x(C(n,0)-xC(n,1)+x^2C(n,2)-...)]'
=[x(1-x)^n]'
=(1-x)^n-x(1-x)^(n-1)
The original formula is exactly the value when x = 1. Obviously, the value is 1 when n = 0 and 0 when n > = 1

How to write 1-1 / 2cn1 + 1 / 3cn2-1 / 4cn3. + (- 1) ^ n 1 / (n + 1) CNN?

Using (n + 1) C (n, K) / (K + 1) = C (n + 1, K + 1)
Original formula = 1 / (n + 1) [C (n + 1,1) - C (n + 1,2) + C (n + 1,3) -. + (- 1) ^ n c (n + 1, N + 1)]
= 1/(n+1) [ 1 - (1-1)^(n+1) ]
= 1/(n+1)

Verification: c0n + 2c1n + 3c2n + +(n+1)Cnn=2n+n•2n-1．

It is proved that s = c0n + 2c1n + 3c2n + +(n + 1) CNN, & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; in reverse order, s = (n + 1) CNN + ncnn-1 + +C0n，∴2S=（n+2）cn0+（n+2）Cn1+… +（n+2）Cnn=（n+2）•2n∴S=2n+n•2n-1．

The distance between a and B is 360 km. The express train and the local train run from the two stations at the same time and meet each other in 3.6 hours. The speed ratio of the express train and the local train is 3:2. How many kilometers does the local train run per hour? How many hours does it take for the express to finish the whole journey?

The speed of express train is: 360 ／ 3.6 × 33 + 2 = 100 × 35 = 60 (km / h), the speed of slow train is: 3600 ／ 3.6-60 = 100-60 = 40 (km / h), 360 ／ 60 = 6 (H). A: slow train travels 40 km per hour, and it takes 6 hours for the express train to complete the whole journey

Fraction: ① 3.04 = () ② 20.4 = () ③ 0.36 = () fraction: ① 5 of 12 = () ② 1 of 5 = () ③ 1 and 8 of 7

Formation fraction:
① 3.04 = (3 and 1 / 25)
② 20.4 = (20 and 2 / 5)
③0.36=（9/25 ）
To decimal:
① 5 / 12 ≈ (0.417)
② 1 / 5 = (0.2)
③ 1 and 8 / 7 ≈ 2.142

A and B depart from AB and travel in opposite directions. A travels 80 kilometers per hour, B travels 10% of the whole journey per hour, and when B reaches five eighths of the whole journey, a will stop
Then go one sixth to B and find out how many kilometers away AB is

"B travels 10% of the whole journey every hour" can be calculated: 1 / 10% = 10 (hours). When B travels to five eighths of the whole journey, it takes 10 * 5 / 8 = 25 / 4 (hours), which is also the time that a has already traveled. When a travels one sixth to B, it means that a has traveled (1-1 / 6) = 4 / 6. (80 * 25 / 4) / (4 / 6) = 900 (km)

The function and relationship of the basic theorem of real number completeness!
I would like to ask six basic theorems of the completeness of real numbers: 1. The principle of truth. 2. Monotone boundedness theorem. 3. Interval nest theorem. 4. Finite cover theorem. 5. Aggregation theorem. 6. Cauchy convergence criterion. What are their functions? It seems that in general mathematical analysis, we directly give theorems and their mutual deduction process

I don't know if I'm right about the six basic theorems about the completeness of real numbers. These six theorems describe a property of real number set from different angles: the real number set is closed about the limit operation, that is, the continuity of real numbers. They are equivalent to each other and can be regarded as axioms. The route to prove the equivalence of seven basic theorems of real numbers is: 1

The two trains leave from two places 798km apart at the same time. After 4.2 hours, the two trains meet. Car a travels 86.7km per hour, and car B how many kilometers per hour?

798 △ 4.2-86.7, = 190-86.7, = 103.3 (km); answer: car B travels 103.3 km per hour

There are six numbers. The average is 8.5. The average of the first four numbers is 9.25. The average of the last three numbers is 10. What is the fourth number?

The total is 8.5x6 = 51
The first four and 9.25x4 = 37
The last three and 3x10 = 30
The fourth number is 37 + 30-51 = 16

If cN0 + 1 / 2cn1 + 1 / 3cn2 +... + 1 / (n + 1) CNN = 31 / (n + 1), find the term with the largest coefficient in the expansion of (1-2x) 2n