Explicit or general rule

The governing concept is an explicit rule for identifying a specified term. The listed terms are x, x + 6, and x + 12, so the third term is x + 12. Since the problem states that this third term equals 25, form the equation x + 12 = 25. Subtracting 12 from both sides gives x = 13, so option B is correct. Substitution verifies the result: when x = 13, the progression becomes 13, 19, 25, 31, …; its third term is exactly 25 and its common difference is 6. If x were 11, 15, or 17, the third term would be 23, 27, or 29 respectively. Thus only option B satisfies the stated condition.

The governing concept is an explicit rule for generating the terms of a sequence. To find the listed progression, substitute successive positive integers for n in aₙ = 5n + 4. For n = 1, a₁ = 5(1) + 4 = 9. For n = 2, a₂ = 5(2) + 4 = 14. For n = 3 and n = 4, the values are 19 and 24. Therefore the progression is 9, 14, 19, 24, ..., which is option C. Its common difference is 5, matching the coefficient of n. Option A begins with 4, option B follows another rule, and option D starts with the second term rather than the first. Thus only C satisfies the rule from n = 1 onward.

An explicit or general rule generates a sequence by substituting successive positive integers for n. For a_n = 7n + 3, put n = 1 to obtain a_1 = 7(1) + 3 = 10; put n = 2 to obtain a_2 = 14 + 3 = 17; then a_3 = 24 and a_4 = 31. Therefore the progression is (10, 17, 24, 31, ...), so option C is correct. The consecutive difference is 7, confirming that it is an arithmetic progression. Option A starts with the constant 3 and omits the first contribution from 7n. Option B lists multiples of 7 and ignores the added 3. Option D begins with 17, which is the second term rather than the first term. Substitution therefore clearly identifies C.

This question uses an explicit or general rule for a geometric progression: aₙ = a₁rⁿ⁻¹. The given rule has first term 81 and common ratio 1/3. To find the fourth term, substitute n = 4. The exponent is n − 1 = 3, because three ratio applications take the first term to the fourth term. Thus a₄ = 81 × (1/3)³ = 81 × 1/27 = 3. Therefore, option B is correct. Option A would use one extra factor of 1/3, while options C and D do not result from multiplying 81 by 1/3 three times. The exponent must be n − 1 rather than n, which is the central point of the calculation.

This question uses an explicit or general rule for a geometric progression. To find the fifth term, substitute n = 5 into aₙ = 64 × (1/2)ⁿ⁻¹. Thus a₅ = 64 × (1/2)⁵⁻¹ = 64 × (1/2)⁴. Since (1/2)⁴ = 1/16, a₅ = 64 × 1/16 = 4. Therefore option B is correct. The exponent is n − 1 because the first term contains zero factors of the common ratio; the fifth term contains four. Using exponent 5 would give 2, which explains option A but violates the stated formula. Three factors give 8, and one factor gives 32, neither of which is a₅.