What do you think the appropriate limit is of this sequence?
0.7, 0.72, 0.727, 0.7272,...
The answer my textbook gives is 27/99. I have no idea how they got that answer. How do I solve this question?
.72*10^0 +.72*10^-2 + .72*10^-4 +.72*10^6
geometric sequence
each term is r = 10^-2 times the preceding term
sum of infinite series
= a/(1-r) = .72/(1-10^-2)
= .72/(1-.01)
= .72/(0.99)
= 72/99
try dividing 72 by 99 on your calculator :)
Ah, the art of finding limits! Let's dive into this sequence of yours and solve the mystery.
To find the limit of the sequence, we need to identify the pattern it follows. In this case, we notice that each term adds another digit to the number 7.
So, let's write out a few more terms to see if the pattern becomes even more apparent:
0.72727, 0.727272, 0.7272727...
From this, we can realize that each term of the sequence is a repeating decimal with 7 as the repeating digit.
Now, if you take a closer look, you might spot that the number of 7s in each term corresponds to the number of digits in that term minus one.
For example, in the first term (0.7), there is one 7. In the second term (0.72), there are two 7s. In the third term (0.727), there are three 7s. And so on…
Following this pattern, we can see that the general term of the sequence can be described as:
0.7 + 0.02/9 + 0.002/9^2 + 0.0002/9^3 + ...
Where 0.02/9 is obtained from adding 2 to the numerator for each term, and 9 is the common ratio.
To find the limit, we can use the formula for the sum of an infinite geometric series:
S = a / (1 - r)
Applying this formula to our sequence, where a = 0.7 and r = 0.1/9, we can calculate:
S = 0.7 / (1 - 0.1/9) = 0.7 / (1 - 0.0111) = 0.7 / 0.9889 ≈ 0.7071...
So, the appropriate limit of your sequence is approximately 0.7071...
And there you have it! Mystery solved, and I hope you found some fun along the way. Keep those mathematical questions coming!
To find the limit of the given sequence, let's analyze the pattern:
0.7, 0.72, 0.727, 0.7272,...
We can observe that each term in the sequence is obtained by adding a 7 at the end, but the number of 7s increases by one each time.
To express this pattern algebraically, let's call the nth term of the sequence "a(n)". We can express a(n) as follows:
a(n) = 0.7 + 0.02 + 0.007 + 0.0002 + ... + (0.007/n)
Now we can rearrange the terms:
a(n) = (7/10) + (2/100) + (7/1000) + (2/10000) + ... + (7/10^n)
Next, we notice that the terms in the sequence form a geometric series with a common ratio of 1/100. The sum of a geometric series with the first term "a" and common ratio "r" can be calculated using the formula:
Sum = a / (1 - r)
In this case, a = (7/10) and r = 1/100. Therefore, the sum of the geometric series is:
Sum = (7/10) / (1 - 1/100) = (7/10) / (99/100) = (7/10) * (100/99) = 70/99
Since the limit of the sequence is the sum of the terms as n approaches infinity, the answer is 70/99.
Therefore, the answer given by your textbook is correct: the appropriate limit of the given sequence is 70/99.
To find the limit of the given sequence, you can look for a pattern in the numbers and determine if there is any convergence towards a certain value.
Looking at the sequence, you can observe that each term is constructed by appending the digit 2 to the previous term. This pattern continues indefinitely.
To find the limit L, you can express the given sequence as an infinite series with a common ratio. Let's call the first term 'a' and the common ratio 'r'. In this case, 'a' would be 0.7 and 'r' would be 0.02 because each term is obtained by adding 0.02 to the previous term.
Now, you can use the formula for the sum of an infinite geometric series to find the limit:
L = a / (1 - r)
Plugging in the values, you get:
L = 0.7 / (1 - 0.02)
L = 0.7 / 0.98
L = 7/10 / 49/50
L = 7/10 * 50/49
L = 350/490
L = 35/49 = 5/7
Hence, the limit of the given sequence is 5/7. It seems like there might be an error in your textbook's answer.
Note: It is a good practice to double-check your answers and consult other reliable sources if you are uncertain about a solution provided in your textbook.