1.calculate the sum of the infinite series S(infinity)=16+4+1+..., (1 point)
A. -16/3
B.64/3
C.12
D. It does not converge
2.Which of the following formulas can be used to write 0.797979... as a fraction?
1. 64/3
2. S(infinity)=(79/100)/(1-(1/100))
1. D. It does not converge.
Don't worry, this series will never reach a finite sum. It will just keep getting smaller and smaller, but will never come to a stop. It's like being stuck in traffic - there's no end in sight!
2. Well, let's look at the repeating decimal 0.797979... It's like a never-ending pattern of 79's, and who doesn't love a good pattern? One way to express this as a fraction is to let x equal 0.797979... and multiply it by 100, so we have 100x = 79.797979... Now, let's subtract the original number x from 100x: 100x - x = 79.797979... - 0.797979... That gives us 99x = 79, and we can solve for x by dividing both sides by 99. So, x = 79/99. And there you have it, 0.797979... as a beautiful fraction!
1. To calculate the sum of the infinite series S(infinity) = 16 + 4 + 1 + ..., we can use the formula for the sum of an infinite geometric series given by S = a / (1 - r), where "a" is the first term and "r" is the common ratio.
In this case, the first term is 16 and the common ratio is 4/16 (obtained by dividing the second term by the first term). Thus, we have a = 16 and r = 4/16 = 1/4.
Plugging these values into the formula, we have S = 16 / (1 - 1/4).
Simplifying further, we get S = 16 / (3/4) = (16 * 4) / 3 = 64 / 3.
Therefore, the sum of the infinite series S(infinity) is 64/3.
Hence, the answer is B. 64/3.
2. To write 0.797979... as a fraction, we can use the formula for a repeating decimal as a fraction. This formula states that for a repeating decimal "x" of the form 0.ababab..., the fraction form is x = a / (99), where "a" is the two-digit repeating portion.
In this case, 0.797979... has the repeating portion "79". Therefore, we can write 0.797979... as 79 / 99.
Therefore, the correct formula to write 0.797979... as a fraction is 79 / 99.
Hence, the answer is D. 79 / 99.
To calculate the sum of the infinite series S(infinity) = 16 + 4 + 1 + ..., we need to determine if the series converges or not.
To check for convergence, we can examine the common ratio of the series. The common ratio is found by dividing any term in the series by its preceding term. In this case, if we divide each term by the previous term, we get:
(16/4) = 4
(4/1) = 4
(1/0.25) = 4
Since the common ratio is the same for all terms and equal to 4, and it is not equal to 1, the series is a geometric series with a common ratio of 4.
Now, to determine if the series converges or diverges, we check if the absolute value of the common ratio is less than 1. In this case, |4| = 4, which is greater than 1. Therefore, the series does not converge.
Hence, the answer to question 1 is D. It does not converge.
Moving on to question 2, we need to find a formula to represent the repeating decimal 0.797979... as a fraction.
One way to approach this problem is to understand that the repeating decimal represents the sum of an infinite geometric series. The repeating part consists of two digits, namely 79. Let's call the repeating part x.
When we write the decimal 0.797979... as a fraction, we can set it up as follows:
0.797979... = x / (100 - 1)
The denominator of the fraction is obtained by subtracting 1 from 100 because each digit in the decimal represents a place value that is a power of 10.
Next, to eliminate the decimal, we multiply both sides of the equation by 100:
100 * 0.797979... = x
79.797979... = x
Now, we subtract the original equation from this equation:
79.797979... - 0.797979... = x - 0.797979...
79 = x - 0.797979...
Simplifying, we find:
79 = x - 0.797979...
Now, let's add the decimal 0.797979... to both sides:
79 + 0.797979... = x
79.797979... = x
We can now substitute the fraction 79/99 for the repeating decimal:
79.797979... = 79/99
Thus, the formula for writing 0.797979... as a fraction is 79/99.
Therefore, the answer to question 2 is: None of the given formulas.
a = 16, r = 1/4
S = a/(1-r) = 16/(1 - 1/4) = 16 * 4/3 = ?
.7979.. = 79/99
or, probably
99x = 79