Prove that x-1 is a factor of x^n-1 for any positive integer n.

is a factor of

Step-by-step explanation:

is a factor of

We will prove this with the help of principal of mathematical induction.

For n = 1, is a factor , which is true.

Let the given statement be true for n = k that is is a factor of .

Thus, can be written equal to  , where y is an integer.

Now, we will prove that the given statement is true for n = k+1

Thus, is divisible by .

Hence, by principle of mathematical induction, the given statement is true for all natural numbers,n.

she gotta workk that a 2

step-by-step explanation:

answer:   the paragraph proof is mentioned below.

step-by-step explanation:

given: in a δxwy where

such that, ∠wzx ≅ ∠yzx and zw≅zy

we have to prove that: segment zx is a perpendicular bisector of segment wy.

prove: since, zw≅zy ⇒ ∠wzy is a supplementary angle.

then, ∠wzy= 180°

⇒∠wzx + ∠yzx = 180° ( because xz divides ∠wzy into two parts)

but, ∠wzx ≅ ∠yzx ( given)

⇒∠wzy+∠wzy=180° ( by substitution)

⇒2∠wzy=180°

⇒∠wzy=90°

⇒∠wzy=∠yzx =90°

therefore, xz⊥wy where wz=zy

⇒zx is a perpendicular bisector of wy.