Cho A = 1/101 + 1/102 + 1/103 + ... + 1/199 + 1/200

To find the value of A, we need to evaluate the sum of the given series.

A = 1/101 + 1/102 + 1/103 + ... + 1/199 + 1/200

To simplify this expression, we can find a common denominator for all the terms. The least common multiple of the numbers from 101 to 200 is 101 * 102 * ... * 199 * 200.

Multiplying both sides of the equation by this common denominator, we get:

A * (101 * 102 * ... * 199 * 200) = (101 * 102 * ... * 199 * 200) * (1/101 + 1/102 + 1/103 + ... + 1/199 + 1/200)

Simplifying the right side of the equation, we have:

A * (101 * 102 * ... * 199 * 200) = (200 * 199 * ... * 103 * 102 * 101 + 200 * 198 * ... * 102 * 101 + ... + 200 * 199 * ... * 104 * 103)

Notice that each term on the right side of the equation has a common factor of 101 * 102 * ... * 199 * 200. Factoring out this common factor, we get:

A * (101 * 102 * ... * 199 * 200) = (101 * 102 * ... * 199 * 200) * (1/200 + 1/198 + ... + 1/102 + 1/100)

Canceling out the common factor on both sides of the equation, we have:

A = 1/200 + 1/198 + ... + 1/102 + 1/100

Now we can evaluate the value of A.

Since 5/8 < a < 3/4, we can find a common denominator for these two fractions, which is 8.

Multiplying both sides of the inequality by 8, we get:

5 < 8a < 6

Dividing both sides of the inequality by 8, we have:

5/8 < a < 6/8

Simplifying, we get:

5/8 < a < 3/4

Therefore, the given inequality holds true.