Explicit or general rule

The governing concept is the nth-term formula for an arithmetic progression: aₙ = a₁ + (n − 1)d. Here the first term is a₁ = 7 and the common difference is d = 12 − 7 = 5. Therefore, aₙ = 7 + (n − 1)5 = 7 + 5n − 5 = 5n + 2. A useful check is to substitute n = 1: 5(1) + 2 = 7, and n = 2 gives 12, so the formula matches the sequence. Option A incorrectly treats 7 as the coefficient of n, option C gives a first term of 3, and option D gives a first term of 7 but has common difference 2 rather than 5. Hence option B is correct.

The governing concept is identifying the position of a term in an explicitly written sequence. The listed terms are first x, second x + 6, and third x + 12. Since the third term is given as 25, set x + 12 = 25. Subtracting 12 from both sides gives x = 13. Substitution verifies the sequence as 13, 19, 25, …, whose third term is indeed 25 and whose common difference is 6. Option A would make the third term 23, option C would make it 27, and option D would make it 29. Therefore, option B is the only value satisfying the condition.

To obtain the sequence from an explicit rule, substitute successive positive integer values of n. For n = 1, a_1 = 6(1) + 1 = 7; for n = 2, a_2 = 13; for n = 3, a_3 = 19; and for n = 4, a_4 = 25. Thus the arithmetic progression begins 7, 13, 19, 25, ... and has common difference 6. Therefore option C is correct. Option A begins with 1 and would correspond to a different rule, while option B begins with 6 and option D begins with the second term 13, so neither gives the required first term. The direct substitution also confirms the answer without needing to guess the pattern.

To find the first positive term, use the condition aₙ > 0. From 6n − 17 > 0, we get 6n > 17, so n > 17/6, which means the smallest positive integer n is 3. Checking the nearby terms confirms this: a₂ = 6(2) − 17 = −5, while a₃ = 6(3) − 17 = 1. Thus the third term is the first positive term, so option B is correct. Since the coefficient of n is positive, the sequence increases as n increases; therefore, once a₃ is positive, later terms are also positive. Options C and D identify positive terms but not the first one, while option A is negative.

An explicit nth-term rule generates the sequence by substituting successive positive integer values of n. For n = 1, a₁ = 5(1) + 4 = 9; for n = 2, a₂ = 10 + 4 = 14; for n = 3, a₃ = 15 + 4 = 19; and for n = 4, a₄ = 20 + 4 = 24. Thus the progression is 9, 14, 19, 24, ..., which is option C. Its common difference is 5, consistent with the coefficient of n. Option A starts with 4, option B corresponds to a different rule, and option D begins with the second term rather than the first. The first term must always be found by putting n = 1.

The governing concept is substitution into an explicit or general rule for a sequence. The rule is a_n = 7n + 4, so calculate each requested term separately. For n = 6, a_6 = 7(6) + 4 = 42 + 4 = 46. For n = 10, a_10 = 7(10) + 4 = 70 + 4 = 74. Adding them gives a_6 + a_10 = 46 + 74 = 120. Hence option C is correct. It is important not to substitute only one index or add the indices first without applying the constant correctly. The nearby alternatives reflect such arithmetic or substitution mistakes; direct evaluation of both terms confirms that 120 is the only correct value.

An explicit or general rule generates a sequence by substituting n = 1, 2, 3, and so on. For a_n = 7n + 3, the first term is a_1 = 7(1) + 3 = 10, the second is a_2 = 7(2) + 3 = 17, the third is 24, and the fourth is 31. Therefore the sequence is (10, 17, 24, 31, ...), so option C is correct. It is also an arithmetic progression because consecutive terms differ by 7. Option A begins with 3, which would result from omitting the 7n part; B starts with 7, and D starts at the second or later value rather than the first term.

The governing concept is an explicit or general rule for a sequence. Each listed term is one less than a power of 2: 1 = 2¹ − 1, 3 = 2² − 1, 7 = 2³ − 1, 15 = 2⁴ − 1, and 31 = 2⁵ − 1. Therefore, when the position is n, the corresponding power is 2ⁿ, so the nth term is aₙ = 2ⁿ − 1. Option A is correct. Option B would begin with 3 rather than 1, option C gives a linear sequence, and option D does not reproduce the given terms.

The governing concept is identifying an explicit rule for a sequence. Test the candidate expressions at n = 1, 2, and 3. The expression n² + 2n + 2 gives 1 + 2 + 2 = 5, 4 + 4 + 2 = 10, so this quick test reveals it does not match the second term; therefore the supplied key and options contain an inconsistency. Checking the sequence carefully, its successive differences are 7, 9, 11, and 13, so the second difference is 2. The actual formula is n² + 4n, which gives 5, 12, 21, 32, and 45. Since that formula is absent from the options, no listed option is correct. This item requires revision rather than a pass.

The governing concept is recognizing an exponential sequence and expressing it explicitly. Rewrite the terms as 8 − 1, 16 − 1, 32 − 1, 64 − 1, and 128 − 1. The powers of 2 are 2³, 2⁴, 2⁵, 2⁶, and 2⁷, so for the nth term the power is 2ⁿ⁺². Therefore aₙ = 2ⁿ⁺² − 1, which is option A. Checking n = 1 gives 2³ − 1 = 7, and n = 5 gives 2⁷ − 1 = 127. Option B is shifted by one power, option C does not preserve the doubling pattern, and option D is linear rather than exponential.

The governing concept is identifying an explicit rule by separating the variable pattern from a constant. The terms can be rewritten as 2 + 2, 4 + 2, 8 + 2, 16 + 2, and 32 + 2. The powers are 2¹, 2², 2³, 2⁴, and 2⁵, so the nth term is aₙ = 2ⁿ + 2. Hence option A is correct. Direct checking gives a₁ = 2 + 2 = 4, a₂ = 4 + 2 = 6, and a₅ = 32 + 2 = 34. Option B gives 4 at n = 1, option C gives 4 at n = 1 but 8 at n = 2, and option D introduces an extra changing n instead of the constant addition.