S=1 + 2(1/5) + 3(1/5)^2 + 4(1/5)^3 +5(1/5)^4…...........infinite......... If
s=a/b
,
where a and
b are coprime positive integers, what is the
value of a+b?
To find the value of S, we can notice that each term is in the form n(1/5)^(n-1), where n represents each consecutive positive integer.
To simplify the expression, let's write out the terms:
S = 1 + 2(1/5) + 3(1/5)^2 + 4(1/5)^3 + 5(1/5)^4 + ...
Now, let's factor out (1/5) from each term:
S = (1/5)[1 + 2(1/5) + 3(1/5)^2 + 4(1/5)^3 + 5(1/5)^4 + ...]
Next, let's consider the sum inside the brackets. This is an infinite geometric series with first term a = 1 and a common ratio r = 1/5.
Using the formula for the sum of an infinite geometric series, we have:
S = (1/5) * [a / (1 - r)]
= (1/5) * [1 / (1 - 1/5)]
= (1/5) * [1 / (4/5)]
= (1/5) * [5/4]
= 1/4
Therefore, S = 1/4.
Since a = 1 and b = 4 are coprime positive integers, the value of a + b is:
a + b = 1 + 4 = 5