Cho biểu thức A:

To find the value of the given expression, we can start by simplifying each term and then adding them together.

The expression can be written as:

A = 1/2.6 + 1/3.8 + 1/4.10 + 1/5.12 + ... + 1/499.1000

Now, let's simplify each term:

1/2.6 = 1/(2*6) = 1/12
1/3.8 = 1/(3*8) = 1/24
1/4.10 = 1/(4*10) = 1/40
1/5.12 = 1/(5*12) = 1/60

We can see that each term is in the form 1/(n*(n+4)), where n is the numerator and n+4 is the denominator.

Now, let's find a pattern for the terms:

1/12, 1/24, 1/40, 1/60, ...

We can observe that the numerator is always 1, and the denominator follows the pattern n*(n+4).

So, the expression can be written as:

A = 1/12 + 1/24 + 1/40 + 1/60 + ... + 1/(499*(499+4))

Now, let's find the common denominator for all the terms:

The common denominator is the least common multiple (LCM) of the denominators, which is 12*24*40*60*...*(499*(499+4)).

To simplify the calculation, we can divide each denominator by 4:

12/4 = 3
24/4 = 6
40/4 = 10
60/4 = 15

So, the common denominator becomes:

3*6*10*15*...*(499*(499+4))

Now, let's simplify the expression:

A = (1/12)*(3/3) + (1/24)*(6/6) + (1/40)*(10/10) + (1/60)*(15/15) + ... + (1/(499*(499+4)))*((499*(499+4))/(499*(499+4)))

A = (3/36) + (6/144) + (10/400) + (15/900) + ... + (499*(499+4))/(499*(499+4))

A = (3 + 6 + 10 + 15 + ... + (499*(499+4)))/(499*(499+4))

Now, let's find the sum of the numerator:

The sum of the first n terms of the series 1, 2, 3, 4, ... is given by the formula n*(n+1)/2.

So, the sum of the first 499 terms is 499*(499+1)/2 = 499*500/2 = 249500.

Now, let's substitute this value into the expression:

A = 249500/(499*(499+4))

A = 249500/(499*503)

A = 249500/250997

A ≈ 0.993