Proof Concerning Central Line X5X6X_5X_6 of Triangle

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Hello everyone. Today we will prove theorem about central line X5X6X_5X_6 of triangle. Proof of the Central Line in a Golden Rectangle Construction Statement of the Theorem Let ABCDABCD be a golden rectangle where ABBC=ϕ \frac{AB}{BC} = \phi , and construct the square BCQP BCQP inside it. Reflect P P over D D to obtain E E . Then, the line EB EB coincides with the central line X5X6 X_5X_6 of triangle ABP ABP . Step-by-Step Proof 1. Define the Coordinates We assign coordinates as follows: A=(0,0) A = (0,0) , B=(ϕx,0) B = (\phi x, 0) , C=(ϕx,x) C = (\phi x, x) , D=(0,x) D = (0, x) . The square BCPQ BCPQ ensures P=(ϕx,2x) P = (\phi x, 2x) . The reflection of P P across D D is E=(−ϕx,2x) E = (-\phi x, 2x) . 2. Compute the Nine-Point Center X5 X_5 The nine-point center X5 X_5 is the circumcenter of the medial triangle, which consists of the midpoints: M1=(0+ϕx2,0)=(ϕx2,0), M_1 = \left(\frac{0 + \phi x}{2}, 0\right) = \left(\frac{\phi x}{2}, 0\right), \[ M_2 = \left(\f...
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Proof of the formula for the sum of the first n Fibonacci numbers

Hello everyone. Today we will prove the formula for the sum of the first nn Fibonacci numbers. We will use the method of mathematical induction as a proof method. 

Theorem: ∀n∈N0,∑j=0nFj=Fn+2−1\forall n \in \mathbb {N} _0, \quad \displaystyle \sum_ {j = 0} ^ nF_j = F_ {n + 2} -1  

Proof: 

1. Base case (n = 0)  ∑j=00Fj=F2−1\displaystyle \sum_ {j = 0} ^ 0F_j = F_ {2} -1 F0=F2−1F_ {0} = F_ {2} -1 0=1−10 = 1-1 0=00 = 0  

2. Induction hypothesis (n = m) 

Suppose that: ∑j=0mFj=Fm+2−1\displaystyle \sum_ {j = 0} ^ mF_j = F_ {m + 2} -1  

3. Inductive step (n = m + 1) 

Using the assumption from the second step, we will prove that: ∑j=0m+1Fj=Fm+3−1\displaystyle \sum_ {j = 0} ^ {m + 1} F_j = F_ {m + 3} -1  So,  ∑j=0m+1Fj=∑j=0mFj+Fm+1=\displaystyle \sum_ { j = 0} ^ {m + 1} F_j = \displaystyle \sum_ {j = 0} ^ {m} F_j + F_ {m + 1} = Fm+2−1+Fm+1=F_ {m + 2} -1 + F_ {m +1} = Fm+1+Fm+2−1=F_ {m + 1} + F_ {m + 2} -1 = Fm+3−1F_ {m + 3} -1

■\blacksquare

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