S=1+2(1/5)+3(1/5)^2+4(1/5)^3….
If S=a/b, where a and b are coprime positive integers, what is the value of a+b?
To find the value of a+b, we need to find the value of a and b first. Then, we can add them together.
Let's try to simplify the expression for S. We can see that each term in the series is of the form n(1/5)^(n-1), where n represents the term number.
Let's rewrite the expression for S using the generic term n(1/5)^(n-1):
S = 1(1/5)^0 + 2(1/5)^1 + 3(1/5)^2 + 4(1/5)^3 + ...
Now, let's rewrite the generic term n(1/5)^(n-1) as n/5^(n-1):
S = (1/5)^0 + (2/5)^1 + (3/5)^2 + (4/5)^3 + ...
We can see that each term in the series can be expressed as (n/5)^(n-1). Let's rewrite the series using this expression.
S = (1/5)^0 + (2/5)^1 + (3/5)^2 + (4/5)^3 + ...
= (1/5)^0 + (2/5)^1 + (3/5)^2 + (4/5)^3 + ...
Now, notice that this series is a geometric series with a common ratio of 1/5. A geometric series can be summed using the formula:
S = a / (1 - r)
where:
- a is the first term of the series, in this case, 1/5^0 = 1
- r is the common ratio of the series, in this case, 1/5
Substituting the values into the formula, we have:
S = 1 / (1 - 1/5)
= 1 / (4/5)
= 1 * (5/4)
= 5/4
Therefore, we have found that S = 5/4. To find the value of a+b, we simply add the numerator (a) and the denominator (b):
a + b = 5 + 4
= 9
Hence, the value of a+b is 9.