Find the value of 2 over 9 minus 4 over 27 plus dot dot dot plus the quotient of the product of negative 1 raised to the n plus 1 power and 2 to the nth power and the quantity 3 raised to the n plus 1 power plus dot dot dot . (10 points)
A) 1/5
B) 1/6
C) 2/15
D) the value of the series is not a finite number
is it 2/15 or 1/6?
a = 2/9
r = -2/3
S = a/(1-r) = (2/9) / (1 + 2/3) = 2/15
If you didn't do the math, you're just guessing.
analy
To find the value of the given series, we need to find a pattern and determine if it converges to a finite number or not. Let's break it down step by step:
The series starts with the fraction 2/9 minus 4/27. We can simplify each fraction:
2/9 - 4/27 = (2*3)/(9*3) - (4*1)/(27*1) = 6/27 - 4/27 = 2/27.
So the series up to this point is 2/27.
Next, we have the quotient of the product of negative 1 raised to the (n+1)th power and 2 to the nth power, and 3 raised to the (n+1)th power.
This part of the series can be written as: (-1)^(n+1) * (2^n) / (3^(n+1)).
As we move forward, we can observe that in each term of the series, the exponent of -1, 2, and 3 increases by 1. This suggests that this is a geometric series.
Now, in a geometric series, the sum of the terms can be found using the formula: a / (1 - r), where "a" is the first term and "r" is the common ratio.
Let's apply this formula to our series. The first term is 2/27, and the common ratio is (-1/2) * (1/3) = -1/6.
The sum of the series, if it converges to a finite number, is given by: (2/27) / (1 - (-1/6)).
Simplifying the denominator, we get: (2/27) / (7/6) = (2/27) * (6/7) = 12/189.
Now, we reduce the fraction: 12/189 = 4/63.
Therefore, the value of the given series is 4/63.
Since none of the answer choices match 4/63, we can conclude that the value of the series is not a finite number. Therefore, the correct answer is option D) the value of the series is not a finite number.