The following equations can be obtained by calculation: 22-12 = 2 × 1 + 1, 32-22 = 2 × 2 + 1, 42-32 = 2 × 3 + 1,..., (n + 1) 2-n2 = 2 × n + 1. Add the above equations to get: (n + 1) 2-12 = 2 × (1 + 2 + 3 +...) +n) 1 + 2 + 3 + +N = n (n + 1) 2 analogy above: please find 12 + 22 + 32 + +The value of N2 (it is required that there must be an operational reasoning process)

The following equations can be obtained by calculation: 22-12 = 2 × 1 + 1, 32-22 = 2 × 2 + 1, 42-32 = 2 × 3 + 1,..., (n + 1) 2-n2 = 2 × n + 1. Add the above equations to get: (n + 1) 2-12 = 2 × (1 + 2 + 3 +...) +n) 1 + 2 + 3 + +N = n (n + 1) 2 analogy above: please find 12 + 22 + 32 + +The value of N2 (it is required that there must be an operational reasoning process)

23-13 = 3 × 12 + 3 × 1 + 1, 33-23 = 3 × 22 + 3 × 2 + 1, 43-33 = 3 × 32 + 3 × 3 + 1.... (n + 1) 3-n3 = 3 × N2 + 3 × n + 1 --- (6 points) add the above formulas to get: (n + 1) 3-13 = 3 × (12 + 22 + 32 +) +n2）+3×（1+2+3… +n) So: 12 + 22 + 32 + +N2 = 13 [(n + 1) 3 − 1 − n − 31 + n2n] = 16N (n + 1) (2n + 1) ---- (12 points)

How many items should be filled in the blank space of 1,1,7,14,6,18,3,12,2, (): A, 10 b, 20 C, 30 d, 40
The calculation process is needed

It should be a, 10
There is no general formula for this sequence, but there are rules to follow
It is not difficult to find that there is a multiple relationship between 1 and 1, 7 and 14, 6 and 18, 3 and 12, and the multiple increases according to the law of 1, 2, 3 and 4. So it is not difficult to infer that the following number should be 5 times of 2, that is, 10, choose a

In the sequence: 1,4,9,16, the 25th number is?

You see:
1 is the square of 1, 1 × 1 = 1, 4 is the square of 2, 2 × 2 = 4, 9 is the square of 3, 3 × 3 = 9, 16 is the square of 4, 4 × 4 = 16······
These numbers are the square multiples of 1, 2, 3 and 4 respectively, and each number is viewed according to the square of the number from small to large, then the 25th number is the square of 25, 25 × 25 = 625
The final answer is 625! I hope I can help you