Sequences

The given nth-term rule is \(a_n=4n+3\). Substituting \(n=8\), we get \(a_8=4\times8+3=32+3=35\). Therefore, 35 is correct. The value 37 would result if the constant added were 5, which is not the case here. Exam tip: For an nth-term question, first substitute the given term number for \(n\), then simplify.

To find the fourth term, substitute \(n=4\) in the rule: \(a_4=3(4)^2+2=3\times16+2=50\). Hence, 50 is correct. The nearby distractor 48 results from stopping after \(3\times16\) and not adding 2. In exams, substitute the value first and then follow the order of operations: power, multiplication, and addition.

Each successive term is 7 greater than the preceding term: 6, 13, 20, 27, 34. Therefore, the term after 20 is 20 + 7 = 27, so 27 is correct. Choosing 28 would make the difference from 20 equal to 8, which breaks the common pattern. Exam tip: Find the difference between consecutive terms to identify the rule of a sequence.

Each term in the sequence is 9 less than the preceding term: \(95-9=86\), and \(86-9=77\). Therefore, the missing value is 86. If 88 were used, the difference from 95 would be 7, so the common pattern would not hold. Exam tip: In a missing-term sequence, compare consecutive known terms first to identify the rule.

The differences between consecutive terms are \(8-3=5\), \(15-8=7\), \(24-15=9\), and \(35-24=11\). These are consecutive odd numbers, so the next difference is \(13\). Therefore, the next term is \(35+13=48\). Choosing 46 would repeat the difference 11, which does not follow the increasing-difference pattern. Exam tip: For such sequences, first list the consecutive differences and look for their pattern.

The successive subtractions are 3, 6, 9, and 12. Since each subtraction increases by 3, the next subtraction is 15. Therefore, the next term is \(90-15=75\). The value 72 would result from subtracting 18 next, which does not follow the given pattern. Exam tip: For such sequences, first find the differences between consecutive terms and then identify the pattern in those differences.

In this sequence, each term is 3 times the preceding term: 5, 15, 45, 135, 405, 1215. Therefore, the sixth term is 1215. The value 405 is the fifth term, so it is a close but incorrect option. Exam tip: In a geometric sequence, check the common ratio between consecutive terms to find the next term.

In this sequence, each term is obtained by dividing the preceding term by 2: \(512\to256\to128\to64\). Therefore, the next term is \(64\div2=32\). Although 16 is related to the pattern, it comes after 32 and is therefore incorrect here. Exam tip: Compare consecutive terms to identify the rule of a sequence.

The terms are \(2^2,3^2,4^2,5^2,6^2,\ldots\). Hence, the \(n\)th term is \((n+1)^2\). Therefore, the \(8\)th term is \((8+1)^2=9^2=81\). Choosing \(64\) is a common error caused by taking \(8^2\), since the first term is \(2^2\), not \(1^2\). Exam tip: always check the offset between the term number and the base being squared.

The given terms are consecutive cubes: \(8=2^3\), \(27=3^3\), \(64=4^3\), and \(125=5^3\). Therefore, the next, or fifth, term is \(6^3=216\). \(225\) is not a cube number, so it cannot be the next term of this sequence. Exam tip: Rewrite sequence terms as cubes to identify the pattern in their bases.

The differences between consecutive terms are \(3,4,5,6\). Therefore, after the fifth term \(20\), the sixth term is \(20+7=27\), and the seventh term is \(27+8=35\). Hence, the correct answer is \(35\). Note that \(27\) is only the sixth term. Exam tip: write the successive differences to identify the next increment in such sequences.

From the third term onward, each term is the sum of the two immediately preceding terms: \(3+4=7\), \(4+7=11\), and \(7+11=18\). Therefore, the next term is \(11+18=29\). Although 27 may result from guessing a pattern in the differences, it does not follow the given sum rule. Exam tip: first check the relation among consecutive terms, then apply the same rule to find the next term.

In this sequence, each term is the sum of the two preceding terms: \(19=7+12\), \(31=12+19\), and \(50=19+31\). Therefore, the next term is \(31+50=81\). Although 79 can be obtained by adding 29 to the previous term, that rule does not fit all the terms. Exam tip: Along with checking differences, test whether each term is formed by adding the previous two terms.

Each term is obtained by multiplying the preceding term by 3: 4, 12, 36, 108, 324. Therefore, 324 is the fifth term. The fourth term is 108, so the fourth-term option is incorrect. Exam tip: List and count the terms in order to identify a term's position.

This is an arithmetic sequence with first term 8 and common difference 7. Its nth term is 8+(n-1)×7. Putting it equal to 57 gives (n-1)×7=49, so n=8. Therefore, 57 is the 8th term. The 7th term is 50, so it is not correct. Exam tip: To find the position of a number in a sequence, equate it to the nth-term expression.

This is a decreasing arithmetic sequence in which each successive term is 6 less. The terms are 88, 82, 76, 70, 64, 58, 52. Therefore, 52 is the 7th term. The 6th term is 58, so option A is not correct. Exam tip: You can list the terms in order or use \(a_n=a+(n-1)d\).

The difference between consecutive terms is 6, so this is an arithmetic progression. Its nth term is \(a_n=a_1+(n-1)d=3+(n-1)\times6=6n-3\). In option B, substituting \(n=1\) gives 9, not the first term 3. Exam tip: always check a proposed rule by substituting \(n=1\).

This is an arithmetic sequence because 9 is added to each term. Here, \(a_1=6\) and the common difference is \(d=9\). Thus, \(a_n=a_1+(n-1)d=6+9(n-1)=9n-3\), so option A is correct. In option B, putting \(n=1\) gives 15 as the first term, so it is incorrect. Exam tip: check an \(a_n\) rule by substituting \(n=1\) to verify the first term.

For the first four terms, substitute n=1, 2, 3, and 4. This gives a_1=5(1)+2=7, a_2=12, a_3=17, and a_4=22. Hence, the correct sequence is (7, 12, 17, 22). Option D incorrectly treats 2 as the first term; it is obtained when n=0, whereas a sequence normally starts with n=1. Exam tip: always check the starting value of n before listing terms.

To find the fifth term, substitute \(n=5\) in the rule: \(a_5=5^2+3\times5=25+15=40\). Therefore, 40 is correct. A value such as 35 can result from an error in calculating \(5^2\) or \(3\times5\). Exam tip: after substituting the term number, evaluate powers first, then multiplication, and finally addition.