a) To find the sum of the series 1/2 + 1/2.3 + 1/3.4 + ... + 1/999.1000, we can write each term as a fraction with a common denominator.

The common denominator for all the terms is 2.3.4...1000, which can be written as 1000!.

So, the series can be written as:

1/(2.3) + 1/(2.3.4) + 1/(3.4.5) + ... + 1/(999.1000)

To simplify the series, we can multiply each term by the missing factors in the denominator:

(1/2) * (4/4) + (1/2.3) * (5/5) + (1/3.4) * (6/6) + ... + (1/999.1000) * (1001/1001)

This gives us:

4/(2.3.4) + 5/(2.3.4.5) + 6/(3.4.5.6) + ... + 1001/(999.1000.1001)

Now, we can see that each term has a common factor in the numerator and denominator, which can be canceled out:

4/(2.3.4) = 1/3
5/(2.3.4.5) = 1/(3.4)
6/(3.4.5.6) = 1/(4.5)

So, the series simplifies to:

1/3 + 1/(3.4) + 1/(4.5) + ... + 1/(999.1000.1001)

Now, we can see that each term has a common factor in the denominator, which can be canceled out:

1/3 = 1/3
1/(3.4) = 1/(3.4)
1/(4.5) = 1/(4.5)

So, the series further simplifies to:

1/3 + 1/(3.4) + 1/(4.5) + ... + 1/(999.1000.1001)

Now, we can see that each term is of the form 1/n(n+1), where n ranges from 3 to 999.

Therefore, the sum of the series is:

1/3 + 1/(3.4) + 1/(4.5) + ... + 1/(999.1000.1001)

b) To find the sum of the series 1/2 - 1/3.4 - 1/4.5 - 1/5.6 + ... + 1/49.50, we can write each term as a fraction with a common denominator.

The common denominator for all the terms is 2.3.4...50, which can be written as 50!.

So, the series can be written as:

1/(2) - 1/(3.4) - 1/(4.5) - ... - 1/(49.50)

To simplify the series, we can multiply each term by the missing factors in the denominator:

(1/2) * (3/3) - (1/3.4) * (5/5) - (1/4.5) * (6/6) - ... - (1/49.50) * (51/51)

This gives us:

3/(2.3) - 5/(3.4.5) - 6/(4.5.6) - ... - 51/(49.50.51)

Now, we can see that each term has a common factor in the numerator and denominator, which can be canceled out:

3/(2.3) = 1/2
5/(3.4.5) = 1/(4.5)
6/(4.5.6) = 1/(5.6)

So, the series simplifies to:

1/2 - 1/(4.5) - 1/(5.6) - ... - 1/(49.50.51)

Now, we can see that each term has a common factor in the denominator, which can be canceled out:

1/2 = 1/2
1/(4.5) = 1/(4.5)
1/(5.6) = 1/(5.6)

So, the series further simplifies to:

1/2 - 1/(4.5) - 1/(5.6) - ... - 1/(49.50.51)

Now, we can see that each term is of the form 1/n(n+1), where n ranges from 2 to 49.

Therefore, the sum of the series is:

1/2 - 1/(4.5) - 1/(5.6) - ... - 1/(49.50.51)