Which expression represents the second partial sum for $\sum_{n=1}^{\infty} 3\left(\frac{1}{2}\right)^{n-1}$ ?
A. $3\left(\frac{1}{2}\right)^0+3\left(\frac{1}{2}\right)^1$
B. $3\left(\frac{1}{2}\right)^1+3\left(\frac{1}{2}\right)^2$
C. $\frac{1}{2}(3)^0+\frac{1}{2}(3)^1$
D. $\frac{1}{2}(3)^1+\frac{1}{2}(3)^2$
Answer (2)
The problem asks for the second partial sum of the series ∑ n = 1 ∞ 3 ( 2 1 ) n − 1 .
The first term is found by setting n = 1 : 3 ( 2 1 ) 1 − 1 = 3 ( 2 1 ) 0 .
The second term is found by setting n = 2 : 3 ( 2 1 ) 2 − 1 = 3 ( 2 1 ) 1 .
The second partial sum is the sum of the first two terms: 3 ( 2 1 ) 0 + 3 ( 2 1 ) 1 , so the answer is 3 ( 2 1 ) 0 + 3 ( 2 1 ) 1 .
Explanation
Understanding the Problem We are asked to find the expression that represents the second partial sum for the series ∑ n = 1 ∞ 3 ( 2 1 ) n − 1 . The second partial sum means we need to add the first two terms of the series.
Finding the First Term The first term of the series is when n = 1 . So, we have 3 ( 2 1 ) 1 − 1 = 3 ( 2 1 ) 0 = 3 ( 1 ) = 3 .
Finding the Second Term The second term of the series is when n = 2 . So, we have 3 ( 2 1 ) 2 − 1 = 3 ( 2 1 ) 1 = 3 ( 2 1 ) = 2 3 .
Calculating the Second Partial Sum The second partial sum is the sum of the first two terms: 3 + 2 3 = 3 ( 2 1 ) 0 + 3 ( 2 1 ) 1 .
Final Answer Therefore, the expression that represents the second partial sum is 3 ( 2 1 ) 0 + 3 ( 2 1 ) 1 .
Examples
Partial sums are useful in many areas of math and physics. For example, when analyzing the behavior of an infinite series, we often look at the sequence of its partial sums to determine if the series converges or diverges. In physics, partial sums can approximate the solution to differential equations or model physical phenomena like the superposition of waves.
The expression representing the second partial sum for the series ∑ n = 1 ∞ 3 ( 2 1 ) n − 1 is 3 ( 2 1 ) 0 + 3 ( 2 1 ) 1 . Therefore, the correct answer is option A.
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