Explicit or general rule

Using \(a_n=3^n-1\), we get \(a_1=3-1=2\), \(a_2=9-1=8\), \(a_3=27-1=26\), and \(a_4=81-1=80\). Hence, option A is correct. Although \(a_n=2^n+n\) may look similar, its first term is 3, so it is incorrect. Exam tip: verify a proposed rule with at least the first three terms.

The general term is \(a_n=3^n-1\). Substituting \(n=4\), we get \(a_4=3^4-1=81-1=80\). Option 81 represents only \(3^4\), but the formula also requires subtracting 1. Exam tip: substitute the term number first, then evaluate powers and remaining operations in order.

(5\cdot3^n) gives (15,45,135,405). In exams check both the base and multiplier in geometric growth.

Given \(a_n=4\cdot2^{n-1}\). Substituting \(n=8\), \(a_8=4\cdot2^{8-1}=4\cdot2^7=4\cdot128=512\). Hence, 512 is the correct option. Getting 256 results from an error in the exponent or multiplication. Exam tip: substitute the value of \(n\) into \(n-1\) first, and then simplify.

\(a_n=2n^2-1\) is a quadratic sequence, so its second differences are constant; here they are 4. In \(5n+2\), the first differences are constant instead. Exam tip: look for an \(n^2\) term.

The governing concept is identifying a quadratic explicit rule. The first differences are 16 − 7 = 9, 31 − 16 = 15, and 52 − 31 = 21. Their second differences are 15 − 9 = 6 and 21 − 15 = 6, which suggests a quadratic expression. Test option C: for n = 1, 3(1)² + 4 = 7; for n = 2, 3(2)² + 4 = 16; for n = 3, 3(3)² + 4 = 31; and for n = 4, 3(4)² + 4 = 52. Hence aₙ = 3n² + 4 and option C is correct. Option A is linear and gives 7, 14, 21, ...; option B gives 7, 10, 15, ...; and option D gives 7, 16, 25, ..., so each fails at a later term.

Given a_n=5n^2+n, a_4=5(4)^2+4=80+4=84 and a_2=5(2)^2+2=20+2=22. Hence, a_4:a_2=84:22=42:11. The ratio 11:42 reverses the terms, while 21:11 simplifies only the first term incorrectly. Exam tip: evaluate both terms separately before reducing a ratio by a common factor.

Given \(a_n=2n^2+4n-3\). Substituting \(n=2\), \(a_2=2(2)^2+4(2)-3=8+8-3=13\). Similarly, for \(n=3\), \(a_3=2(3)^2+4(3)-3=18+12-3=27\). Therefore, option A is correct. In option B, the value of \(a_2\) is incorrect. Exam tip: substitute each value of \(n\) separately and calculate the squared term carefully.

The consecutive differences are \(10,14,18\), and their second differences are \(4,4\); therefore, the sequence has a quadratic general term. Substituting \(n=1,2,3,4\) in \(a_n=2n^2+4n-3\) gives \(3,13,27,45\), respectively. \(a_n=10n-7\) matches only the first two terms, not the later terms. Exam tip: verify a proposed general term using at least the first three terms.

Given \(a_n=n^2+4n\), we get \(a_1=1^2+4(1)=5\), \(a_2=2^2+4(2)=12\), and \(a_3=3^2+4(3)=21\). Therefore, the sum of the first three terms is \(5+12+21=38\). Although 39 is a close distractor, the correct addition gives 38. Exam tip: substitute \(n=1,2,3\) first, then add the resulting terms.

\(a_{n+1}-a_n=[5(n+1)-2]-(5n-2)=5\), which is constant for every term. Hence it is an AP with common difference 5; -2 is only the constant term. Exam tip: in \(pn+q\), \(p\) is the common difference.

Its rule is (a_n=9n-6) and (9n-6=111) gives (n=13). In exams equate the given term to the general term.

Given \(a_n=\frac{3n+2}{4}\), substitute \(n=6\): \(a_6=\frac{3(6)+2}{4}=\frac{18+2}{4}=\frac{20}{4}=5\). The value \(\frac{9}{2}\) may result if the added 2 in the numerator is missed. Exam tip: substitute the given value of \(n\) carefully before simplifying the expression.

The governing concept is forming an explicit rule for a sequence with fractional terms. First express the terms with denominator 4: 5/4, 8/4, 11/4, 14/4, ... . The numerators increase by 3, so the numerator at position n is 3n + 2; therefore aₙ = (3n + 2)/4. Checking gives n = 1: 5/4, n = 2: 8/4 = 2, n = 3: 11/4, and n = 4: 14/4 = 7/2. Thus option B is correct. Option A has numerator increasing only by 1, option C gives a much larger value and the wrong denominator structure, and option D gives integer values rather than the displayed fractions. Converting all terms to a common denominator makes the pattern unambiguous.

The rule for the terms is a_n=11n+2. Thus, a_3=11(3)+2=35 and a_7=11(7)+2=79. Therefore, a_3+a_7=35+79=114. The value 112 may result from omitting the constant term 2 once. Exam tip: substitute the term number correctly for n in each term before adding them.

The direct answer is option B: \(a_n=11n+2\). The differences are \(24-13=11\), \(35-24=11\), and \(46-35=11\), so the sequence increases by 11 each time. Write a linear rule as \(a_n=11n+c\). At the first position, \(11(1)+c=13\), giving \(c=2\). Thus \(a_n=11n+2\), and substitution produces all four displayed terms. Option A, \(13n\), gives 13 first but 26 second, so the increase is wrong. Option B has both the correct difference and correct first term. Option C, \(n+12\), gives 13 first but increases by only 1. Option D, \(11n-2\), increases correctly but gives 9 first, so its constant is wrong. Memory cue: same difference gives the coefficient of \(n\); the first term fixes the added constant.

Given \(a_n=5n^2-2\), \(a_3=5(3)^2-2=45-2=43\) and \(a_4=5(4)^2-2=80-2=78\). Therefore, \(a_3+a_4=43+78=121\). Option 122 could result from an arithmetic error of 1 in a term or in the addition. Exam tip: substitute each value of \(n\) separately and calculate the square first.

The governing concept is recognizing a quadratic sequence from its second differences and verifying its explicit rule. The first differences are 18 − 3 = 15, 43 − 18 = 25, and 78 − 43 = 35. The second differences are 25 − 15 = 10 and 35 − 25 = 10, so a quadratic rule is appropriate. Test option A: when n = 1, 5(1)² − 2 = 3; n = 2 gives 20 − 2 = 18; n = 3 gives 45 − 2 = 43; and n = 4 gives 80 − 2 = 78. Therefore option A is correct. Option B is linear and gives 3, 8, 13, ...; option C gives 3, 12, 27, ...; and option D gives 3, 18, 33, ..., so none reproduces all the terms.

Given \(a_n=2^n+n+1\), substitute \(n=5\): \(a_5=2^5+5+1=32+5+1=38\). Hence, 38 is the correct option. A value of 37 would result from incorrectly omitting the final \(+1\). Exam tip: substitute the value of \(n\) in every part of the rule before simplifying the exponent.

Substituting \(n=1,2,3,4\) in \(a_n=2^n+n+1\) gives \(4,7,12,21\), respectively. Hence, it is the required general term. Although \(a_n=n^2+3\) gives the first three terms \(4,7,12\), its fourth term is \(19\), not \(21\). Exam tip: verify a proposed rule using at least four given terms.

Given (a_n=4^n-2n). Substituting n=3 gives (a_3=4^3-2(3)=64-6=58). Therefore, 58 is the correct answer. An answer such as 56 may result from evaluating 2n incorrectly. Exam tip: while substituting a value in a general term, evaluate the exponent and multiplication carefully.

For option A, substituting \(n=1,2,3,4\) gives \(2,12,58,248\), respectively: \(4^1-2(1)=2\) and \(4^2-2(2)=12\). Hence, the correct explicit rule is \(a_n=4^n-2n\). Option B is only a linear rule, so its second term would be \(6\), not \(12\). Exam tip: test an explicit rule by substituting at least the first three values of \(n\).

Given \(a_n=n^3+2\), set \(a_n=66\): \(n^3+2=66\), so \(n^3=64\). Since \(4^3=64\), \(n=4\) and hence \(a_4=66\). The close distractor \(n=3\) gives \(3^3+2=29\), not 66. Exam tip: subtract the constant first, then identify the cube root.

Substituting \(n=1,2,3,4\) into \(a_n=n^3+2\) gives \(3,10,29,66\), respectively. Hence, the correct general term is \(a_n=n^3+2\). In contrast, \(a_n=n^2+2\) gives \(11\) as the third term, not \(29\). Exam tip: verify a proposed general term by checking at least the first three terms.

For \(a_n=5n-2\), \(a_{n+1}-a_n=[5(n+1)-2]-(5n-2)=5\), which is constant, so it is an AP. For \(n^2+1\), the difference changes. Exam tip: check consecutive-term differences.