The values of 4+(1/4+(1/4+1/4+..upto infinity...
I don't know how to type that fraction .
I have taken a photo and uploaded here.
ibb[dot]co/fLqhxH
Please help me in solving
Your link did not work, but I think you are attempting to write a continued repeating fraction.
Here is a link to introduce you to the topic of continued fractions
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html#section1.5
it is clear that its value > 4 , so let x = 1/4 + [1/4 + [ ..... , only the decimal part
notice that we see that pattern repeating in the second level
SOOO, could we not say: ?
x = 1/(4+x) ??
then, 4x + x^2 = 1
x^2 + 4x - 1 = 0
x = (-4 ± √20)/2
= -2 ± √5, but x is clearly positive,
x = √5 - 2
so your repeating continued fraction = 2 + √5 or appr 4.236068
To solve the given expression, we need to find the sum of the series 4 + (1/4) + (1/4 + 1/4) + (1/4 + 1/4 + 1/4) + ... up to infinity.
Let's call the sum of this series S.
We can observe that each term after the first term is formed by adding an additional 1/4 to the previous term.
Therefore, the second term is 1/4, the third term is 1/4 + 1/4, the fourth term is 1/4 + 1/4 + 1/4, and so on.
We can rewrite this series as:
S = 4 + (1/4) + (1/4 + 1/4) + (1/4 + 1/4 + 1/4) + ...
= 4 + 1/4(1 + 1 + 1 + ...)
The second part of the series, 1 + 1 + 1 + ..., is an infinite geometric series with a common ratio of 1/4.
The sum of an infinite geometric series with a common ratio between -1 and 1 is given by the formula:
Sum = a / (1 - r)
where 'a' is the first term and 'r' is the common ratio.
In this case, a = 1 and r = 1/4.
So, the sum of the second part of the series is:
Sum = 1 / (1 - 1/4)
= 1 / (3/4)
= 4/3
Therefore, the sum of the entire series S is:
S = 4 + 1/4(4/3)
= 4 + 1/3
= 12/3 + 1/3
= 13/3
Hence, the value of the given series is 13/3.
I apologize, but as an AI text-based bot, I am unable to access or view any external images or links. However, I can still assist you in solving the problem you described.
The given series, 4 + (1/4 + (1/4 + 1/4 + ...)), seems to be an infinite series. To find the value of this series, we need to determine the pattern and then calculate the sum.
In this series, it appears that the terms after the first one are all equal to 1/4. So, let's consider the sum of these remaining terms, which can be represented as S.
S = 1/4 + 1/4 + 1/4 + ...
To calculate the sum S, we can use the formula for the sum of an infinite geometric series:
S = a / (1 - r),
where 'a' represents the first term and 'r' represents the common ratio between consecutive terms. In this case, a = 1/4 and r = 1/4.
Substituting the values into the formula:
S = (1/4) / (1 - 1/4).
Simplifying:
S = (1/4) / (3/4).
Invert the denominator and multiply:
S = (1/4) * (4/3) = 1/3.
Therefore, the sum of the series 4 + (1/4 + (1/4 + 1/4 + ...)) is 4 + 1/3 = 13/3.
I hope this helps! Let me know if you have any further questions.