How do you go from the first step where the summation of (1-p)^{k-1} transform to the fraction

hogwartsxhoe5t 2022-10-11 Answered
How do you go from the first step where the summation of ( 1 p ) k 1 transform to the fraction
k = 1 n p ( 1 p ) k 1 = p 1 ( 1 p ) n 1 ( 1 p )
How to simplify the summation?
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Answers (2)

Teagan Zamora
Answered 2022-10-12 Author has 18 answers
Explanation:
For all a 1, one can prove that k = 0 n a k = 1 a n + 1 1 a , for all n N
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djo57bgfrqn
Answered 2022-10-13 Author has 4 answers
Step 1
Firstly, lets get rid of p,which appears everywhere:
k = 1 n p ( 1 p ) k 1 = p k = 1 n ( 1 p ) k 1
Now,lets denote a = 1 p,where a is an integer.
Now denote with S = k = 1 n a k 1
Lets calculate the product between a and S:
a S = k = 1 n a k
a S S = k = 1 n a k S
Now lets subtract S from the sum:
( a 1 ) S = k = 1 n a k k = 1 n a k 1
( a 1 ) S = a n 1
S = a n 1 a 1
S = 1 a n 1 a
Step 2
Now all that we have to do is to replace a with p 1
S = 1 ( 1 p ) n 1 ( 1 p )
p S = p 1 ( 1 p ) n 1 ( 1 p )
p 1 ( 1 p ) n 1 ( 1 p ) = p S = p k = 1 n ( 1 p ) k 1 = k = 1 n p ( 1 p ) k 1
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