Explicit or general rule

For \(a_n=3n^3-n\), substituting \(n=1,2,3,4\) gives \(2,22,78,188\), respectively, so it is the correct general term. The close distractor \(2n^3\) gives the first term as 2, but for \(n=2\) it gives 16, not 22. Exam tip: test at least the first two or three values of \(n\) when checking a proposed general term.

(a_4=67) and (a_8=39) so the sum is (106). In exams find both terms carefully in a decreasing formula.

At (n=1) it gives (88) and at (n=2) it gives (81) so (a_n=95-7n). In exams check the first two terms of a decreasing sequence.

Putting \(n=4\), \(a_4=\frac{4(3\times4+2)}{4}=\frac{4(14)}{4}=14\). Hence, the correct answer is \(14\). Option \(13\) may result from calculating \(3n+2\) incorrectly. Exam tip: substitute the value of \(n\) first and evaluate the expression inside the brackets carefully.

The governing concept is verifying an explicit formula for a sequence, including fractional values. Test option A by substituting the first four positive integers. For n = 1, a₁ = 1(3 + 2)/4 = 5/4. For n = 2, a₂ = 2(6 + 2)/4 = 16/4 = 4. For n = 3, a₃ = 3(9 + 2)/4 = 33/4. For n = 4, a₄ = 4(12 + 2)/4 = 14. Thus option A reproduces every displayed term and is correct. Option B produces 1, 3, 6, 10, while option C produces 1, 4, 7, 10. Option D produces 1, 3, 19/3, and 11, so these alternatives fail to match the sequence. Direct substitution is the safest check when the terms are fractional and quadratic.

To find the third term, substitute n=3 in the rule: \(a_3=5^3-3=125-3=122\). Therefore, 122 is correct. The value 125 is only \(5^3\); 3 must also be subtracted. Exam tip: Substitute the term number into the general rule first, then evaluate powers and remaining operations in order.

The first five terms are (7,19,31,43,55) and the average is (31). In exams divide the sum by the number of terms.

Its rule is (a_n=12n-5) and (12n-5=151) gives (n=13). In exams equate the given term to the general term.

The consecutive differences are \(8,10,12\), and their second differences are constant at \(2\), so the general term should be quadratic. Substituting \(n=1,2,3,4\) in \(a_n=n^2+5n+4\) gives \(10,18,28,40\), respectively. Option A matches only the first two terms; at \(n=3\), it gives \(26\), not \(28\). Exam tip: always test a proposed general term using at least the first three terms.

The rule is \(a_n=4n^2-2n+3\). Substituting \(n=5\), \(a_5=4(5)^2-2(5)+3=4\times25-10+3=93\). Therefore, 93 is correct. A value such as 95 usually results from an error while evaluating the \(-2n\) term. Exam tip: substitute the value of \(n\) into each term separately before simplifying.

The governing concept is finding the index of a specified term from a linear explicit rule. Set the expression equal to the required value: 8n − 3 = 77. Adding 3 to both sides gives 8n = 80, and dividing by 8 gives n = 10. Therefore 77 occurs at the tenth term, so option B is correct. Substitution confirms the result: a₁₀ = 8(10) − 3 = 80 − 3 = 77. The distractors correspond to nearby indices, but they give different values: a₉ = 69, a₁₁ = 85, and a₁₂ = 93. The essential distinction is between the term value, 77, and its position, n = 10. Writing the equation before solving prevents confusion and ensures that the index is obtained rather than another sequence value.

The governing concept is generating sequence terms from an explicit formula. Numbering normally begins with n = 1, so evaluate the rule at n = 1, 2, 3, and 4. For n = 1, a₁ = 1² + 5(1) + 6 = 1 + 5 + 6 = 12. For n = 2, a₂ = 2² + 10 + 6 = 20. For n = 3, a₃ = 9 + 15 + 6 = 30. For n = 4, a₄ = 16 + 20 + 6 = 42. Hence the first four terms are 12, 20, 30, 42, so option C is correct. The other choices result from changing the constant, using an incorrect starting index, or making arithmetic errors. It is important to substitute each index carefully and square n before multiplying by 5.

Use the explicit rule at n = 4 because a₄ is given. Substitution gives a₄ = 4² + 4d + 2 = 16 + 4d + 2 = 18 + 4d. Setting this equal to 34 gives 18 + 4d = 34, so 4d = 16 and d = 4. Therefore, option C is correct. The other options result from omitting the constant 2 or making an error while solving the linear equation.

Answer: B, 64. The formula gives a term when the corresponding value of n is substituted. For a4, put n = 4: a4 = 2(4²) + 5(4) − 3 = 2(16) + 20 − 3 = 32 + 20 − 3 = 49. For a2, put n = 2: a2 = 2(2²) + 5(2) − 3 = 2(4) + 10 − 3 = 8 + 10 − 3 = 15. Therefore a4 + a2 = 49 + 15 = 64. Option A, 51, does not equal the calculated sum. Option C, 57, and option D, 60, are also not the result of adding 49 and 15. A common mistake is to calculate a4 correctly but accidentally use the wrong value for a2 or forget the square on n. Memory cue: substitute the index carefully, simplify powers first, and add only after both terms are known.

Answer: A, aₙ = 2n² + 4n − 4. Find the first differences: 12 − 2 = 10, 26 − 12 = 14, and 44 − 26 = 18. The differences increase by 4, so a quadratic expression is appropriate. Test option A. At n = 1, it gives 2 + 4 − 4 = 2. At n = 2, it gives 8 + 8 − 4 = 12. At n = 3, it gives 18 + 12 − 4 = 26. At n = 4, it gives 32 + 16 − 4 = 44. Therefore it matches all four terms. Option B gives 6 at n = 1, so it fails immediately. Option C is linear and gives 2, 12, 22, 32, not the sequence. Option D gives 2, 14, 28, 44 and fails at the second and third terms. Memory cue: constant second differences indicate a quadratic rule; always test the formula against several terms.

The governing concept is evaluating an explicit rule at two positions and then simplifying the resulting ratio. For n = 5, a₅ = 5³ + 5² = 125 + 25 = 150. For n = 3, a₃ = 3³ + 3² = 27 + 9 = 36. Thus a₅ : a₃ = 150 : 36. The greatest common divisor of 150 and 36 is 6, so dividing both parts by 6 gives 25 : 6. Therefore option B is the simplified answer. Option A is the correct unsimplified ratio, but the question asks for the ratio in standard simplified form. Option C reverses the order to a₃ : a₅, and option D incorrectly uses only selected powers instead of evaluating the whole formula at both indices.

The governing concept is evaluating an explicit sequence rule at two specified indices and subtracting the results in the stated order. Substitute n = 4 first: a_4 = 4(4^2) + 3(4) + 2 = 4(16) + 12 + 2 = 78. Next substitute n = 1: a_1 = 4(1^2) + 3(1) + 2 = 4 + 3 + 2 = 9. Therefore, a_4 − a_1 = 78 − 9 = 69, so option B is correct. Option A can arise from an error while evaluating the quadratic term, whereas options C and D may result from incorrect substitution or subtraction. Computing the two terms separately is useful because it prevents confusion between the indices and preserves the required subtraction order.

The governing concept is generating terms from an explicit formula by substituting successive positive integer values of n. For the first four terms, use n = 1, 2, 3, and 4. At n = 1, a_1 = 1^2 + 6(1) + 5 = 1 + 6 + 5 = 12. At n = 2, a_2 = 4 + 12 + 5 = 21. At n = 3, a_3 = 9 + 18 + 5 = 32. At n = 4, a_4 = 16 + 24 + 5 = 45. Thus the sequence begins 12, 21, 32, 45, making option A correct. Option B starts with the result obtained by omitting the constant correctly or miscalculating the expression; options C and D similarly reflect arithmetic or indexing errors. The index should begin at 1 for this sequence.

The governing concept is direct substitution into an explicit sequence rule written in product form. To obtain the first four terms, use n = 1, 2, 3, and 4. We get a_1 = 1(1 + 3) = 4, a_2 = 2(2 + 3) = 10, a_3 = 3(3 + 3) = 18, and a_4 = 4(4 + 3) = 28. Therefore, the first four terms are 4, 10, 18, and 28, so option B is correct. Option A begins with the value obtained by using n = 0 and then does not follow the required starting index. Option C follows n(n + 1), not n(n + 3), while option D does not consistently satisfy the multiplication rule. Keeping both factors visible makes the substitutions easier to check.

The governing concept is converting the first term and common difference of an arithmetic sequence into an explicit formula. The standard rule is a_n = a_1 + (n − 1)d. Here a_1 = 17, and because each next term is 9 less, the common difference is d = −9. Therefore, a_n = 17 + (n − 1)(−9) = 17 − 9n + 9 = 26 − 9n. Substitution confirms the formula: for n = 1, a_1 = 26 − 9 = 17; for n = 2, a_2 = 26 − 18 = 8, which is exactly 9 less. Option A gives 8 as the first term because it omits the n − 1 adjustment. Option C increases, and option D has both the wrong direction and growth pattern. Thus B is correct.

The governing concept is evaluating an explicit rule at successive positive integer indices while keeping the exponential and linear parts separate. For n = 1, a_1 = 2^1 + 3(1) = 2 + 3 = 5. For n = 2, a_2 = 2^2 + 3(2) = 4 + 6 = 10. For n = 3, a_3 = 2^3 + 3(3) = 8 + 9 = 17. For n = 4, a_4 = 2^4 + 3(4) = 16 + 12 = 28. Therefore, the first four terms are (5, 10, 17, 28), so option A is correct. Option B underestimates the contribution of the linear part, while C and D introduce incorrect additions at later positions. The exponent applies only to the base 2; it does not apply to 3n or to the entire sum. Writing each component separately prevents that common error.

The governing concept is evaluating a formula at two different indices and then adding the resulting terms. First calculate a_4: a_4 = 6(4^2) - 1 = 6(16) - 1 = 96 - 1 = 95. Next calculate a_2: a_2 = 6(2^2) - 1 = 6(4) - 1 = 24 - 1 = 23. Therefore, a_4 + a_2 = 95 + 23 = 118. Option A is correct. A likely error is to omit the -1 twice or to add the indices before applying the formula; those mistakes can produce distractor values such as 120 or other nearby numbers. Each term must be evaluated separately because the expression asks for the sum of two sequence terms, not the value of the rule at n = 6.

Because the fifth term is known, substitute n = 5 into the explicit formula. We obtain a₅ = 5² + 5q + 4 = 25 + 5q + 4 = 29 + 5q. Equating this to 54 gives 29 + 5q = 54, so 5q = 25 and q = 5. Hence option C is correct. The other choices arise from an incorrect substitution of 5 or from mishandling the constant term.

Direct answer: option C, n = 20. The formula gives the value of the term at position n. To find which position has value 155, set the formula equal to 155: 8n − 5 = 155. Add 5 to both sides, obtaining 8n = 160. Divide both sides by 8, so n = 20. A direct check gives a₂₀ = 8(20) − 5 = 160 − 5 = 155. Thus the twentieth term is 155. Option A gives 8(18) − 5 = 139, so it is too small. Option B gives 152 − 5 = 147, also too small. Option D gives 168 − 5 = 163, which is too large. Only C satisfies the original equation. The important idea is that n is the unknown position, so we solve an equation rather than simply substitute the answer choices without a plan. Memory cue: desired term value equals the rule; isolate n step by step.

The governing concept is an explicit or general rule: a single formula must give the term at every positive integer position n. Substitute the first four positions into option A. For n=1, it gives 1(5−1)/2=2. For n=2, it gives 2(10−1)/2=9. For n=3, it gives 3(15−1)/2=21. For n=4, it gives 4(20−1)/2=38. Therefore option A exactly reproduces the sequence. Option B produces 3, 7, 12, and 18, so it fails immediately. Option C produces 2, 9, 16, and 23, matching only the first two terms. Option D produces 2, 8, 18, and 32, so it also fails. Checking several positions establishes that A is the only unambiguous answer.