1/11.x-8/11.15-8/15.19-8/19.23 -...- 8/51.55 =2/55

To find the value of the expression, we need to simplify the given expression:

1/11 - 8/11 + 15/19 - 8/15 + 19/23 - 8/19 + 23/27 - ... - 8/51 + 55

We can see that the terms in the expression follow a pattern:

1/11 - 8/11 = -7/11
15/19 - 8/15 = 97/285
19/23 - 8/19 = 249/437
23/27 - 8/23 = 437/621
...

We can continue this pattern until we reach the term 8/51.

Now, let's calculate the value of the expression:

-7/11 + 97/285 + 249/437 + 437/621 + ... + 8/51 + 55

To simplify this expression, we need to find a common denominator for all the fractions. The least common multiple of the denominators 11, 285, 437, 621, ..., 51 is 11 * 285 * 437 * 621 * ... * 51.

Let's denote this common denominator as D.

Now, we can rewrite the expression with the common denominator:

(-7 * D)/(11 * D) + (97 * D)/(285 * D) + (249 * D)/(437 * D) + (437 * D)/(621 * D) + ... + (8 * D)/(51 * D) + (55 * D)/(D)

Notice that D/D = 1, so we can simplify the expression further:

(-7 * D + 97 * D + 249 * D + 437 * D + ... + 8 * D + 55 * D) / D

Now, we can combine the numerators:

(-7 + 97 + 249 + 437 + ... + 8 + 55) * D / D

The numerator is an arithmetic series, and we can find its sum using the formula:

Sum = (n/2)(first term + last term)

In this case, the first term is -7 and the last term is 55. The number of terms, n, can be found by subtracting the numerator of the last term (51) from the numerator of the first term (-7) and adding 1:

n = (-7 - 51) + 1 = -57 + 1 = -56

Now, we can substitute these values into the formula for the sum:

Sum = (-56/2)(-7 + 55)
= (-28)(-7 + 55)
= (-28)(48)
= -1344

Therefore, the value of the expression is:

-1344 * D / D

Since D/D = 1, the value of the expression is -1344.