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NNADVOKAT [17]
3 years ago
12

Prove that the bisector of the vertical angle of isosceles triangle is the perpendicular bisector of the base

Mathematics
1 answer:
julia-pushkina [17]3 years ago
7 0

Answer:

We will take two angles and name them 'x'

And their sum must equal 180

So,

x + x = 180

2x = 180

x = 180/90

x = 90

Hence, it is proved that the bisector of the vertical angle of isosceles triangle is the perpendicular bisector of the base.

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Solve the following recurrence relation: <br> <img src="https://tex.z-dn.net/?f=A_%7Bn%7D%3Da_%7Bn-1%7D%2Bn%3B%20a_%7B1%7D%20%3D
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a_n = a_{n-1} + n

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a_{n-2} = a_{n-3} + (n - 2) \implies a_n = a_{n-3} + n + (n - 1) + (n - 2)

and the pattern continues down to the first term a_1=0,

a_n = a_{n - (n - 1)} + n + (n - 1) + (n - 2) + \cdots + (n - (n - 2))

\implies a_n = a_1 + \displaystyle \sum_{k=0}^{n-2} (n - k)

\implies a_n = \displaystyle n \sum_{k=0}^{n-2} 1 - \sum_{k=0}^{n-2} k

Recall the formulas

\displaystyle \sum_{n=1}^N 1 = N

\displaystyle \sum_{n=1}^N n = \frac{N(N+1)}2

It follows that

a_n = n (n - 2) - \dfrac{(n-2)(n-1)}2

\implies a_n = \dfrac12 n^2 + \dfrac12 n - 1

\implies \boxed{a_n = \dfrac{(n+2)(n-1)}2}

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Step-by-step explanation:

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