Supposerisarealnumberotherthan1.Proveusingmathematicalinductionthatforeverynon1.46.Supposerisarealnumberotherthan1.Proveusingmathematicalinductionthatforeverynonnegativeintegern,E（上面是n下面i=0）r的i次方=

Question

Suppose r is a real number other than 1.Prove using mathematical induction that for every non
1.46.Suppose r is a real number other than 1.Prove using mathematical
induction that for every nonnegative integer n,
E （上面是n 下面i=0）r的i次方=（1-（r的n+1次方））/1-r

Answer 1

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