Prove that sum_(i=1)^n ((i),(2))=((n+1),(3)) for all n>=2

e1s2kat26 2020-11-07 Answered

Prove that i=1n(i2)=(n+13)
n2

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Expert Answer

crocolylec
Answered 2020-11-08 Author has 100 answers

Step 1. Definitions
Definition combination
nCr=(nr)=n!r!(nr)!
with n!=n×(n1)××2×1.
Pascal's equation
(n+1k)=(nk)+(nk1)
Step 2. Solution
To proof: i=1n(i2)=(n+1) for all positive integers.
Proof by induction
Let P(n) be the statement "i=1n(i2)=(n+1)".
Basis step. n=2
i=1n(i2)=i=12(i2)=(i2)+(22)=0+1=1
(n+13)=(2+13)=(33)=1
Thus P(2) is true.
Inductive step. Let P(k) be true.
i=1k(i2)=(k+13)
We need to proof that P(k+1) is true.
i=1k+1(i2)
=i=1k+1(i2)+(k+12)
(k+13)+(k+12) Since P(k) is true
=((k+1)+13) Pascal's identity
Thus P(k+1) is true.
Conclution. By the principle of methematical induction, P(n) is true for all positive integers n.

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