Explicit or general rule

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 first term is (2) and the common difference is (7), so (a_n=2+(n-1)7=7n-5). In a linear sequence the coefficient of (n) is the common difference.

The first term is (31) and the difference is (-5), so (a_n=31+(n-1)(-5)=36-5n). In a decreasing sequence take the common difference as negative.

Given \(a_n=3n^2+2n\), substitute \(n=4\) to find the fourth term: \(a_4=3(4)^2+2(4)=3\times16+8=48+8=56\). Therefore, 56 is correct. The option 54 may result from an error while squaring or multiplying. Exam tip: In expressions with powers, evaluate \(n^2\) first, then multiply and add.

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.

The numerator is (2n+1) and the denominator is (3n+1), so (a_n=\frac{2n+1}{3n+1}). In fractions identify the numerator and denominator rules separately.

The governing concept is substitution in an explicit or general rule. The rule gives each term directly from its position n, so use n = 1, 2, 3, and 4 in order. For n = 1, a₁ = 2(1)(1 + 1) = 4; for n = 2, a₂ = 2(2)(3) = 12; for n = 3, a₃ = 2(3)(4) = 24; and for n = 4, a₄ = 2(4)(5) = 40. Therefore option B is correct. Option A begins with values obtained from a different expression, while options C and D also shift the substitution and do not satisfy the given rule.

For the fourth term, substitute \(n=4\): \(a_4=7\cdot3^{4-1}=7\cdot3^3=7\cdot27=189\). Therefore, 189 is correct. The value 63 may result from incorrectly using \(7\cdot3^2\). Exam tip: In a general-term formula, substitute the term number first and simplify the exponent carefully.

This is (4^2,5^2,6^2,7^2,\ldots), so (a_n=(n+3)^2). In square sequences relate the base number to (n).

(a_5=13) and (b_3=19), so the sum is (32). With two rules the term numbers may differ, so use them carefully.

The rule (a_n=2n^2+3n+3) gives (a_9=192), so the shown options do not match. This is an option-consistency check question.

Substituting \(n=1,2,3,4\) into \(a_n=n^2+3n\) gives \(4,10,18,28\), respectively. Therefore, option A is correct. Option C gives the first two terms as \(4,10\), but its third term is \(20\), not \(18\). Exam tip: Verify a proposed sequence rule using at least three or four terms.

This is an arithmetic progression with first term 11 and common difference 7. Its general term is \(a_n=11+(n-1)\times7=7n+4\). Setting \(7n+4=74\) gives \(n=10\), so 74 is the tenth term. The ninth term is \(7(9)+4=67\), not 74. Exam tip: To find the position of a number in an AP, equate it to \(a_n\) and solve for \(n\).

Given \(a_n=4^n-2\), substitute \(n=3\): \(a_3=4^3-2=64-2=62\). Hence, \(62\) is correct. The close distractor \(64\) is only the value of \(4^3\); the subtraction of \(2\) must still be done. Exam tip: substitute the term number first, then evaluate the exponent before subtracting.

The numerator is (3n+1) and the denominator is (2n+3), so (a_n=\frac{3n+1}{2n+3}). In a fractional sequence form rules for both parts separately.

(a_7-a_3=16) and there are four gaps, so (d=4), then (a_n=4n). From two given terms first find the common difference.

For the first three terms, substitute \(n=1,2,3\). Thus, \(a_1=2(1)^2-1+4=5\), \(a_2=2(2)^2-2+4=10\), and \(a_3=2(3)^2-3+4=19\). Therefore, the correct sequence is \((5, 10, 19)\). Option C cannot be correct because its first term is 6. Exam tip: when a general term is given, begin with \(n=1\) and calculate each term systematically.

Given (a_n=mn-4), put n=6 to get (a_6=6m-4). Thus, (6m-4=50), so (6m=54) and (m=9). Hence, 9 is the correct option. For example, if m=8, then (a_6=44), not 50. Exam tip: When a particular term is given, substitute its index in the general-term rule.