Evaluate โ
(1
2)๐
โ
๐=1
A. 2
B. 1
C. 0.5.
D. 0.25
We can begin by writing out the terms of the series:
(1/2)^1 + (1/2)^2 + (1/2)^3 + ...
Notice that this is a geometric series with first term 1/2 and common ratio 1/2. Thus, we can use the formula for the sum of an infinite geometric series:
โ(1/2)^r = a/(1 - r)
where a is the first term and r is the common ratio. Plugging in a = 1/2 and r = 1/2, we get:
โ(1/2)^r = (1/2)/(1 - 1/2) = 1
Therefore, the sum of the series is 1.
Answer: B. 1
To evaluate the series, let's break it down step-by-step:
Step 1: Rewrite the series as a sum.
โ(1/2)^r, where r goes from 1 to infinity.
Step 2: Substitute the first few terms to look for a pattern.
When r = 1, the term is (1/2)^1 = 1/2.
When r = 2, the term is (1/2)^2 = 1/4.
When r = 3, the term is (1/2)^3 = 1/8.
As we can see, each term is getting smaller and smaller.
Step 3: Determine the general form of the series.
From the patterns above, the general form of each term is given by (1/2)^r, where r is the position of the term.
Step 4: Determine the sum of the series.
To find the sum of an infinite geometric series, we can use the formula:
S = a / (1 - r),
where S is the sum of the series, a is the first term, and r is the common ratio.
In this case, a = 1/2 and r = 1/2, since each term is multiplied by 1/2 to get the next term.
Plugging in the values into the formula:
S = (1/2) / (1 - 1/2) = (1/2) / (1/2) = 1.
Therefore, the sum of the given series is 1.
Therefore, the correct answer is B. 1.