Find the pattern there must be an explanation Find the pattern, talk about it in detail

1^3 + 2^3 + n^3 = 1/4 [n(n+1)]^2 =1/4(50x51)^2

The sum formula for the sum of powers is n 0 n + 1

The sum formula for the sum of 1 power is n 1 n(n+1) 2

The sum formula for the sum of the 2nd power is n 2 n(n+1)(2n+1) 6

Take the formula: (x+1) 4-x 4=4*x 3+6*x 2+4*x+1

The coefficient can be determined by the Yang Hui triangle.

Then there is: n+1) 4-n 4=4n 3+6n 2+4n+1....1)

n^4-(n-1)^4=4(n-1)^3+6(n-1)^2+4(n-1)+1...2)

n-1)^4-(n-2)^4=4(n-2)^3+6(n-2)^2+4(n-2)+1...3)

2^4-1^4=4×1^3+6×1^2+4×1+1...n)

So (1) + (2) + (3) +n) Yes.

Left = (n+1) 4-1

Right = 4 (1 3 + 2 3 + 3 3 + .n^3)+6(1^2+2^2+3^2+..n^2)+4(1+2+3+..n)+n

So, let's take the three formulas that have been proven.

4(1^3+2^3+3^3+..n^3)+6(1^2+2^2+3^2+..n^2)+4(1+2+3+..n)+n=(n+1)^4-1

Get 4 (1 3 + 2 3 + 3 3 +...n^3)+n(n+1)(2n+1)+2n(n+1)+n=n^4+4n^3+6n^2+4n

After moving the item, we get 1 3+2 3+3 3+...n^3=1/4 (n^4+4n^3+6n^2+4n-n-2n^2-2n-2n^3-3n^2-n)

After merging the same terms on the right side of the equal sign, we get 1 3+2 3+3 3+...n^3=1/4 (n^4+2n^3+n^2)

i.e. 1 3+2 3+3 3+...n^3= 1/4 [n(n+1)]^2

Succeed with flying colors! The cube sum formula is derived.

1^3+2^3+3^3+..n^3= 1/4 [n(n+1)]^2

The result is 55 squared i.e. 3025

1 = 1 1 squared 1

1 +2 = 9 (1 2) squared 9

1 +2 +3 = 36 (1 2 3) 361 +2 +3 +4 = 100 (1 2 3 4) squared 1001 +2 +3 +4 ......10³= ＝（1＋2＋3＋..10) The square of 55 is 3025

Question 1: An apple is next to a peach, so fill in the peach at the horizontal line.

In the second question, a cylinder is next to a rectangle, so fill in the cylinder at the horizontal line.

In the third question, two pineapples are next to an apple, so fill in the pineapple at the horizontal line.

1. The initials of Chinese characters containing the radicals of the character "Zhu" are all "zh" of the tongue, and the finals are also "u"2, the initials of the Chinese characters containing the radicals of the word "main" are the "zh" of the tongue, and the finals are also "u"3, and the initials of the Chinese characters on the radicals of the character "Shang" are the "sh" of the tongue, and the finals are all post-nasal "ang"4, and the initials of the Chinese characters containing the radicals of the word "cheng" are the "ch" of the tongue, and the finals are all post-nasal "eng"5, containing "中" The initials of the Chinese characters in the radicals of the word are all "zh" of the tongue, and the finals are all post-nasal "ong", and the time is relatively tight, so you can refer to it first It may not be standard

Satisfied. See the three numbers in the middle position below the cake.

It was observed that 26 in the third row is equal to 8+18 in the fourth row.

From this, it is found that the law is that the upper number is the sum of the lower two numbers.

From left to right, the first empty space in the third row is 24, and the second empty space is 24.

The first empty space in the second row is 50, and the second empty space is 50.

The first row is empty at 100.

The last one of the numerators is 3 larger than the previous one, the odd term of the denominator is the square of the number of terms, which is rotten plus one, and the even term is the square of the number of terms minus one.

The next number of dividing Zheng yards is 21 + 3 = 24, and the sub-cong and the mother is 8 squared minus one is 63

This question seems to be a bit of a skill to deal with;

This official examination question;

Let's take a look at the numbers:

77 23 and a group; 77+23=100=10^2;

63 18 and a group; 63+18=81=9^2;

23 41 and a trace of slag group; 23+41=64=8^2;

18 31 and a group; 18+31=49=7^2;

Presents the law of decreasing squares; 10^2 9^2 8^2 7^2……；

Thus: 41 and the number of parentheses () and a group; 41+（）36=6^2；

Hence this number: 36-41=-5;

Hope it helps! Happy studying!!

The intervals add up to the number of squares. Fill -5

20 The first place is 1*10+10+0 and the last place in the circle.

28 The first place is 1*10+10+8 and the last place in the circle.

65 The first 5*10+10+5 is the last place in the upper circle.

87 The first 7*10+10+7 is the last place in the upper circle.

There is another kind. The first 1*3+2*3+8*3-1-2-8=20, the second 1*3+8*3+5*3-1-8-5=28, so the third 5*3+4*3+7*3-5-4-7=32, and the fourth is x= of 5*3+7*3+3x=87+5+7+x

!!!I didn't make it!!

There are a few ways to do it, but you can't use all three numbers.

(3)108

The rightmost column is 9 times the number in the middle column.

The bottom row is 3 times the number in the middle row.

x adds 2 each time

x=18

(3) is 108, and the law is that the third row is 4 times that of the first.

4) It's 18, and the rule is +2 each time in the same position