Eight natural numbers are arranged in a row. Starting from the third number, each number is the sum of the two numbers in front of it. Given that the fifth number is 7, then the eighth number is 7______ ．

Eight natural numbers are arranged in a row. Starting from the third number, each number is the sum of the two numbers in front of it. Given that the fifth number is 7, then the eighth number is 7______ ．

Suppose the first number is x, the second number is y, then the third number is x + y, the fourth number is x + 2Y, and the fifth number is 2x + 3Y, that is, 2x + 3Y = 7; because X and y are natural numbers, so x = 2, y = 1; the eighth number is 8x + 13y = 8 × 2 + 13 = 29, so the answer is 29

Natural numbers 1,2,3 What is the sum of all natural numbers?

1+9999=10000
2+9998=10000
…………
4999+5001=10000
Results = 4999 * 10000 + 5000 = 49995000

Proof: for any natural number n, 1 * 2 * 3... * k + 2 * 3 * 4... (K + 1) +... N (n + 1)... (n + k-1) = [n (n + 1)... (n + k)] / (K + 1)
Using mathematical induction

Verification: 1 * 2 * 3 *... * k + 2 * 3 * 4 *... * (K + 1) +... + n (n + 1) * *(n + k-1) = [n (n + 1) *... * (n + k)] / (K + 1) (n is a natural number) proof 1: mathematical induction. Omitted. Proof 2: split term method. 1 * 2 * 3... * k = (- 0 * 1 * 2 * 3... * k + 1 * 2 * 3... * k * (K + 1)) / (K + 1) 2 * 3... * k * (K + 1) = (- 1 * 2 * 3...)