Express the repeating decimal as a fraction (by using infinite series):
0.451141414 . . .
0.451141414..
= 451/1000 + sum(k=0..∞) 14/1000 * 0.01^k
solve for me
To express the repeating decimal 0.451141414... as a fraction using an infinite series, we can set up an equation.
Let x = 0.451141414...
We can multiply both sides of the equation by 10000 to remove the decimal places:
10000x = 4511.4141414...
Next, we subtract the original equation from the multiplied equation to eliminate the repeating part:
10000x - x = 4511.4141414... - 0.451141414...
Simplifying the right side:
9999x = 4511.4141414...
We can remove the repeating part by subtracting the original equation from this new equation:
9999x - x = 4511.4141414... - 0.451141414...
Simplifying the right side:
9998x = 4511.414
Now we can solve for x by dividing both sides by 9998:
x = 4511.414 / 9998
Finally, simplifying the fraction:
x = 2257.707 / 4999
Thus, the repeating decimal 0.451141414... can be expressed as the fraction 2257.707 / 4999.
To express the repeating decimal 0.451141414... as a fraction using an infinite series, we need to identify a pattern in the repeating part.
Let's denote the repeating part as 'x' for convenience. From the given decimal, we observe that the repeating part 0.1414... appears twice. Therefore, we can express it as:
x = 0.1414...
To eliminate the repeating part, we can multiply the equation by a power of 10 that moves the decimal point to the right of the repeating part. In this case, multiplying by 100 (10^2) will work:
100x = 14.1414...
Now, we subtract the original equation from the multiplied equation to eliminate the repeating part:
100x - x = 14.1414... - 0.1414...
Simplifying:
99x = 14
To obtain the fraction form, we divide both sides of the equation by 99:
x = 14/99
Hence, the repeating decimal 0.451141414... can be expressed as the fraction 14/99.