Explicit or general rule

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 first term is (13) and the difference is (11) so (a_n=11n+2). In exams find the constant part from the first term.

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.

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.