Identify the correct mathematical statement(s): A. The expression 3/4 + 5/36 +…

Concept: Four standard simplification tools are combined here — telescoping a fraction series by splitting each term into a difference of reciprocal squares, combining rational expressions over a common denominator (using 1 − x2 = −(x − 1)(x + 1)), applying × and ÷ strictly before + and − (left to right), and factorising a difference of squares a2 − b2 = (a − b)(a + b).

1. Statement A: each term has the form (2n + 1)/[n2(n + 1)2], for n = 1 up to n = 9 (n = 1 gives 3/4, n = 9 gives 19/8100). Since (n + 1)2 − n2 = 2n + 1, each term equals 1/n2 − 1/(n + 1)2, so the whole sum telescopes to 1/12 − 1/102 = 1 − 0.01 = 0.99. This is not 0.09, so Statement A is false.

2. Statement B: write x/(x − 1) + 1/(x + 1) + 2x/(1 − x2) over the common denominator (x − 1)(x + 1), using 1 − x2 = −(x − 1)(x + 1) for the third term. The numerator becomes x(x + 1) + (x − 1) − 2x = x2 + x + x − 1 − 2x = x2 − 1, so the expression reduces to (x2 − 1)/(x2 − 1) = 1 for every x ≠ ±1. This matches the claimed value of 1, so Statement B is true.

3. Statement C: applying × and ÷ before + and − (left to right within each group): 4.2 ÷ 0.7 = 6, then 6 × 3 = 18; and 5 × 2 = 10, then 10 ÷ 0.5 = 20. The expression becomes 3.8 − 18 + 20 = 5.8, not the claimed 5.6, so Statement C is false.

4. Statement D: [(3.63)2 − (2.37)2]/(3.63 + 2.37) is a difference of squares divided by the sum of the same two numbers, so it reduces to (3.63 − 2.37)(3.63 + 2.37)/(3.63 + 2.37) = 3.63 − 2.37 = 1.26, not the claimed 6, so Statement D is false.

Cross-check: Statement B holds for any permitted x, so substituting x = 2 independently verifies it — 2/1 + 1/3 + 4/(1 − 4) = 2 + 0.333… − 1.333… = 1, matching the algebra above. Statement A’s telescoping pattern checks out the same way on a shorter run: the first two terms alone, 3/4 + 5/36, sum to 1/12 − 1/32 = 8/9 ≈ 0.889, consistent with the general pattern.

Since only Statement B holds, the correct selection is B only.