What is the greatest common factor of 100 + n ^ 2 and 2n + 1?

What is the greatest common factor of 100 + n ^ 2 and 2n + 1?

First of all, if 100 + n ^ 2 > 2n + 1, then the greatest common factor should be 2n + 1, that is, 100 + n ^ 2 is a multiple of 2n + 1. If n = 200 is solved, the greatest common factor is 401

-1+3-5+...+（-1）^n(2n-1)=(-1)^n X n
1. (changed to Σ = - 1) ^ n * n)
(1) When n = 1, Σ = - 1 = - 1) ^ 1 * 1
The equation holds
(2) When n = 2, Σ = - 1 + 3 = 2 = - 1) ^ 2 * 2
The equation holds
(3) If n = k, the equation holds
∑＝(-1)^k*k+(-1)^(k+1)*[2(k+1)-1]＝-(-1)^(k+1)*k+(-1)^(k+1)*(2k+1)＝(-1)^(k+1)*(k+1)
That is, the equation n = K + 1 holds
(4) According to mathematical induction, for n ∈ n, there are - 1 + 3-5 +... (- 1) ^ n * (2n-1) = (- 1) ^ n * n

Σ means sum;
The third step is n = K + 1, because it has been assumed that n = k is true, then the sum of n = K + 1 is the sum of n = k, plus (- 1) ^ (K + 1) * [2 (K + 1) - 1]. After simplification, it is found that it is the value of n = K + 1 on the right, and the equation holds when n = K + 1
This is called mathematical induction, which is very useful in proving such problems

Extract the common factor solution a ^ 3 + 1 4x ^ 4-13x ^ 2 + 9 x ^ 3 + 9 + 3x ^ 2 + 3x

a³+1=（a+1）（a²-a+1）
4x^4-13x^2+9=(4x²-9）（x²-1）=（2x+3)(2x-3)(x+1)(x-1)
x^3+9+3x^2+3x=x²（x+3）+3（x+3）=（x²+3)(x+3)