Explicit or general rule

The governing concept is evaluating an explicit sequence rule at the correct index. In the usual indexing of a sequence, the first term is obtained by setting n = 1, not n = 0. Therefore a₁ = 30 − 4(1) = 30 − 4 = 26, so option B is correct. The rule also has common difference −4, since aₙ₊₁ − aₙ = [30 − 4(n + 1)] − [30 − 4n] = −4, confirming that it describes an arithmetic progression. Option A is the constant term in the formula, not the first sequence value. Option C would result from incorrectly using n = 1.5, while option D does not follow from the stated rule.

The governing concept is the explicit or general term of an arithmetic progression, a_n = a + (n − 1)d. The first term is a = 4, and the common difference is d = 11 − 4 = 7. Substitution gives a_n = 4 + (n − 1)7 = 4 + 7n − 7 = 7n − 3. Therefore option A is correct. A quick check is useful: for n = 1, option A gives 7(1) − 3 = 4, and for n = 2 it gives 14 − 3 = 11, matching the sequence. Option B gives 11 for the first term, option C gives 11, and option D gives 10; hence they fail the initial-term test even before later terms are considered.

The governing concept is solving an explicit sequence rule for the position n. We need the term whose value is 49, and the rule is aₙ=6n+1. Set the expression equal to the required value: 6n+1=49. Subtract 1 from both sides to obtain 6n=48, then divide by 6 to get n=8. Therefore 49 occurs at the eighth term, so option C is correct. A direct check gives a₈=6(8)+1=48+1=49. The other choices correspond to n=6, 7, and 9, which give 37, 43, and 55 respectively. Solving the equation is more reliable than listing many terms and clearly establishes the unique position.

The governing concept is an explicit rule, which gives each term directly from its position. To find the first four terms, substitute n = 1, 2, 3, and 4 into a_n = 4n + 3. Thus a_1 = 4(1) + 3 = 7, a_2 = 4(2) + 3 = 11, a_3 = 4(3) + 3 = 15, and a_4 = 4(4) + 3 = 19. Therefore, option A is correct. Option C starts with the value for n = 0, whereas the first term conventionally uses n = 1. Option B omits the added 3, and option D begins with an incorrect value. The constant increase of 4 between the correct terms is also consistent with the formula.

The governing concept is an explicit or general rule for a sequence. Such a rule gives the value of a term directly from its position n. To find the position at which the sequence equals 85, set the rule equal to 85: 9n + 4 = 85. Subtracting 4 from both sides gives 9n = 81, and dividing by 9 gives n = 9. Therefore, 85 occurs at the ninth term, so option C is correct. Verification is direct: a₉ = 9(9) + 4 = 81 + 4 = 85. The alternatives do not satisfy the equation: n = 7 gives 67, n = 8 gives 76, and n = 10 gives 94. Thus only the ninth position produces the required value.

To generate a sequence from an explicit or general rule, substitute successive positive integer values of n. For n = 1, a₁ = 6(1) + 1 = 7. For n = 2, a₂ = 6(2) + 1 = 13; for n = 3, a₃ = 19; and for n = 4, a₄ = 25. Hence the progression begins 7, 13, 19, 25, … and has common difference 6, which is option C. Option A begins at 1 and does not match the value at n = 1. Option B begins at 6, although the formula gives 7 for the first term. Option D begins with 13, which is actually the second term of the required progression. Direct substitution establishes both the starting value and the successive difference, so there is no ambiguity.

A term is positive when aₙ > 0. Using the given rule, 6n − 17 > 0, so 6n > 17 and n > 17/6. Since 17/6 lies between 2 and 3, the smallest positive integer value of n is 3. Direct checking confirms this: a₂ = 6(2) − 17 = −5, whereas a₃ = 6(3) − 17 = 1. Thus the third term is the first positive term, so option B is correct. Because the coefficient of n is positive, the sequence increases by 6 each time; consequently, later terms remain positive after the third term. Options C and D are positive-term positions but not the first, while option A is still negative.

The governing concept is substitution into an explicit general rule for a sequence. The rule is aₙ = 7n + 4, so evaluate each requested term separately. For n = 6, a₆ = 7(6) + 4 = 42 + 4 = 46. For n = 10, a₁₀ = 7(10) + 4 = 70 + 4 = 74. Their sum is therefore a₆ + a₁₀ = 46 + 74 = 120, making option C correct. It is not correct to use only one index or to add the indices first and mishandle the constant. As a check, the rule increases by 7 for each increase of one in n, and the two direct evaluations are consistent. The other choices result from arithmetic or substitution errors.

Direct answer: the nth term is n(n + 1) = n² + n, so A is correct. Look at the terms as products of consecutive integers: 2 = 1×2, 6 = 2×3, 12 = 3×4, and 20 = 4×5. The position number n is the first factor, and the next integer n + 1 is the second factor. Therefore aₙ = n(n + 1) = n² + n. Check each given position: n = 1 gives 1×2 = 2, n = 2 gives 2×3 = 6, n = 3 gives 3×4 = 12, and n = 4 gives 4×5 = 20. B produces 4, 6, 8, so it fails at the first term and does not reproduce the pattern. C produces 3, 6, 11, and D produces 2, 5, 8; neither matches all terms. A useful observation is that the differences are 4, 6, 8, which increase, so this is not a simple arithmetic sequence.

Direct answer: a₄ + a₆ = 114, so C is correct. The formula gives any term when its index n is substituted. For the fourth term, put n = 4: a₄ = 11(4) + 2 = 44 + 2 = 46. For the sixth term, put n = 6: a₆ = 11(6) + 2 = 66 + 2 = 68. Add the two values: a₄ + a₆ = 46 + 68 = 114. The calculation can also be shortened: (11×4 + 2) + (11×6 + 2) = 11(10) + 4 = 114. A, 106, and D, 118, result from arithmetic mistakes or mishandling the constants. B, 110, is what one might obtain by using only 11(4 + 6) and forgetting that the constant 2 appears in both terms. Important reminder: the expression +2 is included once for every evaluated term, so it contributes 4 to the sum.

Direct answer: option A, 2ⁿ − 1. To find an explicit rule, compare each term with powers of 2. The first term is 1 = 2¹ − 1, the second is 3 = 2² − 1, the third is 7 = 2³ − 1, the fourth is 15 = 2⁴ − 1, and the fifth is 31 = 2⁵ − 1. Thus the exponent matches the position number n, so aₙ = 2ⁿ − 1. Checking n = 1 gives 1, n = 2 gives 3, and n = 5 gives 31. Option B is wrong because it uses one extra power and gives 3 for the first term. Option C is a linear rule and gives 1, 3, 5 rather than 1, 3, 7. Option D gives 0 for n = 1 and does not reproduce the sequence. Memory cue: write the terms as powers of 2 minus 1; the exponent tells you the position.

Direct answer: option C, 6561. Each term is obtained by multiplying the previous term by 3, so this is a geometric sequence with first term a = 3 and common ratio r = 3. The nth-term formula is aₙ = arⁿ⁻¹. For the eighth term, a₈ = 3 × 3⁷ = 3⁸ = 6561. The exponent is 7 in the multiplication form because moving from the first term to the eighth term requires seven multiplications; equivalently, 3 × 3⁷ becomes 3⁸. Option A, 2187, is 3⁷ and misses one factor of 3. Option B, 729, is 3⁶ and misses two factors. Option D, 243, is 3⁵ and misses three factors. A useful check is to continue: 81, 243, 729, 2187, 6561; the eighth term is indeed 6561. Memory cue: for a geometric sequence, count the number of jumps from the first term, which is n − 1.

Direct answer: option A, 2ⁿ + 2. Separate each term into a power of 2 plus a constant: 4 = 2² + 2, 6 = 2³? No; instead observe the powers 2, 4, 8, 16, 32 and add 2: 4 = 2¹ + 2, 6 = 2² + 2, 10 = 2³ + 2, 18 = 2⁴ + 2, and 34 = 2⁵ + 2. Therefore the nth term is aₙ = 2ⁿ + 2. Check: n = 1 gives 4, n = 2 gives 6, and n = 5 gives 34. Option B gives 4 at n = 1 but gives 5 at n = 2, so it only matches accidentally at the beginning. Option C gives 4 at n = 1 but 8 at n = 2, not 6. Option D gives 4 at n = 1 and 6 at n = 2, but gives 11 at n = 3 instead of 10 because n is not a constant. The key is that the difference from the powers of 2 is always 2.

The governing concept is identifying an explicit exponential pattern. Rewrite every given term by subtracting 3: 13 + 3 = 16 = 2⁴, 29 + 3 = 32 = 2⁵, 61 + 3 = 64 = 2⁶, 125 + 3 = 128 = 2⁷, and 253 + 3 = 256 = 2⁸. The exponent begins at 4 when n = 1 and increases by 1, so it is n + 3. Consequently, aₙ = 2ⁿ⁺³ − 3, which is option B. Checking n = 1 gives 2⁴ − 3 = 13 and n = 5 gives 2⁸ − 3 = 253. Option A has both an incorrect exponent and constant, option C fails at the first term, and option D describes linear rather than exponential growth.

Direct answer: option A, −1. Evaluate the formula separately at n = 6 and n = 7. For n = 6, the exponent is even, so (−1)⁶ = 1. Hence a₆ = 1(6 + 2) = 8. For n = 7, the exponent is odd, so (−1)⁷ = −1. Hence a₇ = −1(7 + 2) = −9. Add the signed values: a₆ + a₇ = 8 + (−9) = −1. Option B, 1, results from reversing the final sign or subtracting in the wrong order. Option C, −17, incorrectly adds the magnitudes with a negative sign. Option D, 17, adds 8 and 9 while ignoring the negative sign of a₇. The key idea is that (−1)ⁿ changes sign according to parity: it is positive for even n and negative for odd n. Memory cue: even index means plus, odd index means minus; apply the sign before adding.