what is happening mathematically with this...
0.1
0.10
0.101
0.1010
0.10101
well, the terms are getting larger.
If you carry on forever, the sun is 10/99.
The numbers you provided appear to be binary fractions. In binary representation, each digit after the binary point represents a decreasing power of 2 starting from 1/2, then 1/4, 1/8, and so on.
Let's break down each number step-by-step:
1. 0.1: This represents 1/2 in binary, which is equivalent to 0.5 in decimal.
2. 0.10: Here, we have 1/2 + 0/4, so the value is still 0.5.
3. 0.101: Now, we have 1/2 + 0/4 + 1/8, which equals 0.625 in decimal.
4. 0.1010: This represents 1/2 + 0/4 + 1/8 + 0/16, resulting in 0.625 again.
5. 0.10101: Finally, we have 1/2 + 0/4 + 1/8 + 0/16 + 1/32, which is equal to 0.65625 in decimal.
So, the numbers you provided represent the decimal values 0.5, 0.5, 0.625, 0.625, and 0.65625 respectively, in increasing precision.
It seems that you are interested in understanding the pattern behind a sequence of numbers: 0.1, 0.10, 0.101, 0.1010, and 0.10101. Let's break it down and see what's happening mathematically.
Each number in the sequence has digits after the decimal point, with the first number having one digit and each subsequent number adding an additional digit after the decimal point. The digits themselves follow a particular pattern: 0, 1, 0, 1, 0.
If we analyze the pattern, we can see that the sequence alternates between 0 and 1. The first digit after the decimal point is 0, then it switches to 1, back to 0, and so on. Each number in the sequence includes all the previous digits and appends the next digit according to the pattern.
In terms of mathematical representation, we can express the sequence using a recursive formula. Let's denote the nth number in the sequence as S(n):
S(1) = 0.1
S(n) = S(n-1) + 0.1^(n), for n > 1
So, the nth number is obtained by taking the previous number in the sequence, S(n-1), and adding 0.1 raised to the power of n. This recursive formula ensures that each number incorporates all the preceding digits and adds the next digit following the given pattern.
To find any specific term in this sequence, one can either manually calculate it using the recursive formula or use a programming language or calculator to automate the process.