Suppose r is a real number other than 1.Prove using mathemat
1个回答
Check n=0
Left Hand Side=r^0=1
Right Hand Side=(1-r)/(1-r)=1=Left Hand Side
Suppose n=k
We have (k)Σ(i=0)r^i=(1-r^(k+1))/(1-r)
As n=k+1
(k+1)Σ(i=0)r^i
=r^(k+1)+(k)Σ(i=0)r^i
=r^(k+1)+(1-r^(k+1))/(1-r)
=[(1-r)r^(k+1)+1-r^(k+1)]/(1-r)
=[1-r*r^(k+1)]/(1-r)
=[1-r^[(k+1)+1]]/(1-r)
which is the Right Hand Side with (n=k+1)
By Mathematical Induction,
we conclude that
(n)Σ(i=0)r^i=(1-r^(n+1))/(1-r)
is true for all integer n>=0
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