To solve this equation, we need to find the value of X that satisfies the equation:

2/5x8 + 2/8x11 + 2/11x14 + ... + 2/X(X+31) = 101/770

To simplify the equation, let's find a common denominator for all the fractions on the left side:

The common denominator for 5, 8, 11, ..., X is 5 * 8 * 11 * ... * X.

Multiplying each term by the common denominator, we get:

(2 * (8 * 11 * ... * X))/(5 * 8 * 11 * ... * X) + (2 * (5 * 11 * ... * X))/(8 * 11 * ... * X) + (2 * (5 * 8 * ... * X))/(11 * ... * X) + ... + (2 * (5 * 8 * 11 * ... * (X-1)))/(X * (X+31)) = 101/770

Simplifying the fractions, we have:

(2 * (8 * 11 * ... * X))/(5 * 8 * 11 * ... * X) + (2 * (5 * 11 * ... * X))/(8 * 11 * ... * X) + (2 * (5 * 8 * ... * X))/(11 * ... * X) + ... + (2 * (5 * 8 * 11 * ... * (X-1)))/(X * (X+31)) = 101/770

Canceling out the common factors in each term, we get:

2/5 + 2/8 + 2/11 + ... + 2/(X * (X+31)) = 101/770

Now, let's find a common denominator for all the fractions on the left side:

The common denominator for 5, 8, 11, ..., X, and (X * (X+31)) is 5 * 8 * 11 * ... * X * (X+31).

Multiplying each term by the common denominator, we get:

(2 * (8 * 11 * ... * X * (X+31)))/(5 * 8 * 11 * ... * X * (X+31)) + (2 * (5 * 11 * ... * X * (X+31)))/(8 * 11 * ... * X * (X+31)) + (2 * (5 * 8 * ... * X * (X+31)))/(11 * ... * X * (X+31)) + ... + (2 * (5 * 8 * 11 * ... * (X-1) * (X+31)))/(X * (X+31)) = 101/770

Simplifying the fractions, we have:

(2 * (8 * 11 * ... * X * (X+31)))/(5 * 8 * 11 * ... * X * (X+31)) + (2 * (5 * 11 * ... * X * (X+31)))/(8 * 11 * ... * X * (X+31)) + (2 * (5 * 8 * ... * X * (X+31)))/(11 * ... * X * (X+31)) + ... + (2 * (5 * 8 * 11 * ... * (X-1) * (X+31)))/(X * (X+31)) = 101/770

Canceling out the common factors in each term, we get:

2/(5 * (X+31)) + 2/(8 * (X+31)) + 2/(11 * (X+31)) + ... + 2/(X * (X+31)) = 101/770

Now, let's find a common denominator for all the fractions on the left side:

The common denominator for (5 * (X+31)), (8 * (X+31)), (11 * (X+31)), ..., and (X * (X+31)) is (5 * 8 * 11 * ... * X * (X+31)).

Multiplying each term by the common denominator, we get:

(2 * (5 * 8 * 11 * ... * X))/(5 * 8 * 11 * ... * X * (X+31)) + (2 * (5 * 8 * 11 * ... * X))/(8 * 11 * ... * X * (X+31)) + (2 * (5 * 8 * 11 * ... * X))/(11 * ... * X * (X+31)) + ... + (2 * (5 * 8 * 11 * ... * (X-1)))/(X * (X+31)) = 101/770

Canceling out the common factors in each term, we get:

2/(X+31) + 2/(X+31) + 2/(X+31) + ... + 2/(X * (X+31)) = 101/770

Since there are X terms on the left side, we can rewrite the equation as:

2/(X+31) * X = 101/770

Simplifying, we have:

2X/(X+31) = 101/770

Cross-multiplying, we get:

1540X = 101(X+31)

Expanding, we have:

1540X = 101X + 3131

Subtracting 101X from both sides, we get:

1439X = 3131

Dividing both sides by 1439, we get:

X = 3131/1439

Therefore, the value of X that satisfies the equation is approximately 2.17.