K= 2/13 + 2/3.5 + 2/5.7 +...+ 2 99.101

To find the value of K, we need to sum up the given series:

K = 2/13 + 2/3.5 + 2/5.7 + ... + 2/99.101

To simplify the series, we can write it as:

K = 2(1/13 + 1/3.5 + 1/5.7 + ... + 1/99.101)

Now, let's find a pattern in the series:

1/13 = 1/(2 + 11) = 1/2 - 1/11
1/3.5 = 1/(2 + 1.5) = 1/2 - 1/1.5
1/5.7 = 1/(2 + 3.7) = 1/2 - 1/3.7

We can observe that each term in the series can be written as the difference between two fractions.

Using this pattern, we can rewrite the series as:

K = 2[(1/2 - 1/11) + (1/2 - 1/1.5) + (1/2 - 1/3.7) + ... + (1/2 - 1/99.101)]

Now, we can simplify the series further:

K = 2[(1/2 + 1/2 + 1/2 + ... + 1/2) - (1/11 + 1/1.5 + 1/3.7 + ... + 1/99.101)]

The first part of the series simplifies to:

(1/2 + 1/2 + 1/2 + ... + 1/2) = (99)(1/2) = 99/2

Now, let's simplify the second part of the series:

(1/11 + 1/1.5 + 1/3.7 + ... + 1/99.101)

To find a common denominator, we can multiply each fraction by the product of all the denominators:

(1/11 + 1/1.5 + 1/3.7 + ... + 1/99.101) = (1/11)(1.5)(3.7)...(99.101) + (1/1.5)(11)(3.7)...(99.101) + ... + (1/99.101)(11)(1.5)(3.7)...(99)

Now, we can add up all the fractions:

(1/11)(1.5)(3.7)...(99.101) + (1/1.5)(11)(3.7)...(99.101) + ... + (1/99.101)(11)(1.5)(3.7)...(99)

This is a telescoping series, which means that most of the terms will cancel out when we add them up. Only the first and last terms will remain:

(1/11)(1.5)(3.7)...(99.101) + (1/99.101)(11)(1.5)(3.7)...(99)

Now, we can simplify this expression:

(1/11)(1.5)(3.7)...(99.101) + (1/99.101)(11)(1.5)(3.7)...(99)
= (1/11)(1.5)(3.7)...(99.101) + (1/11)(1.5)(3.7)...(99)

Notice that the second term is the same as the first term, except for the last factor of 99.101. Therefore, we can combine these two terms:

(1/11)(1.5)(3.7)...(99.101) + (1/11)(1.5)(3.7)...(99) = (1/11)(1.5)(3.7)...(99.101 + 99)

Now, we can simplify further:

(1/11)(1.5)(3.7)...(99.101 + 99) = (1/11)(1.5)(3.7)...(99.101 + 99)
= (1/11)(1.5)(3.7)...(198.101)

Finally, we can substitute this simplified expression back into the original equation for K:

K = 2[(1/2 + 1/2 + 1/2 + ... + 1/2) - (1/11 + 1/1.5 + 1/3.7 + ... + 1/99.101)]
= 2[(99/2) - (1/11)(1.5)(3.7)...(198.101)]

Therefore, the value of K is 2[(99/2) - (1/11)(1.5)(3.7)...(198.101)].