N is an integer, simplify the root n (n + 1) (n + 2) (n + 3) + 1. And use the result to find the value of the root 2000 times 2001 times 2002 times 2003 plus 1

N is an integer, simplify the root n (n + 1) (n + 2) (n + 3) + 1. And use the result to find the value of the root 2000 times 2001 times 2002 times 2003 plus 1

√[n(n + 1)(n + 2)(n + 3)+ 1]
= √[(n² + 3n)(n² + 3n + 2) + 1]
= √[(n² + 3n)² + 2(n² + 3n) + 1]
= √ (n² + 3n + 1)²
= n² + 3n + 1
So √ (2000 × 2001 × 2002 × 2003 + 1)
= 2000² + 3×2000 + 1
= 4000000 + 6000 + 1
= 4006001

1. If the integer n satisfies (N-2000) ^ 2 + (2001-n) ^ 2 = 1, find the value of n
also:
2. If (a + b) ^ 2 = 7, (a-b) ^ 2 = 3, find the value of a ^ 2 + B ^ 2 and ab
3. Let A-B = - 3, find the value of (a ^ 2 + B ^ 2) / 2-ab

As shown in the figure, in the triangle ABC, ∠ C = 90 °, point D is a point on BC, de ⊥ AB is in E, DC = De, ∠ CAD = ∠ B. verification: AC = half ab

Proof: ∵ de ⊥ ab
∴∠C=∠AED=90°
In RT △ ACD and RT △ AED
AD=AD
DC=DE
∴RT△ACD≌RT△AED（HL）
∴∠CAD=∠BAD
∵,∠CAD=∠B
∴∠BAD=∠B
∵∠C=90°
∴∠CAB+∠B=90°
∴3∠B=90°
∠B=30°
In a right triangle, the corresponding side with an angle of 30 ° equals half of the hypotenuse