pls i need help to proof the following by induction technique
(summation) from i=0 to n for x^i= [(1-x^(n+1)]/ [1-x]
where x does not equal to 1
first i substituted the letter i by a
then i thought about adding 2 (since 1 is not accepted) by didnt go anywhere
Try to prove that P(k) being true causes P(k+1) to be true
If P(k) is true, this will be
Hence P(k) true causes P(k+1) to be true
Therefore if the formula is valid for i=0, it's valid for i=1 and therefore for i=2,
for i=3, for i=4, for i=5........ to infinity
To check the validity for i=0
Hence the hypothesis is valid for x not equal to 1
x cannot be 1 as 1-x in the denominator would be zero,
and we need to avoid having a divide by zero.
Choosing i=1 would be testing the formula for the first 2 terms in the sum,
though you may start at i=0, which strictly speaking is more correct.
If the sum was from i=1 to n, we test the formula for i=1,
but since the sum starts at i=0, then i=0 gives the first term in the sum.