Sequences

To find the fourth term, substitute \(n=4\) in \(a_n=3n+2\): \(a_4=3\times4+2=12+2=14\). Hence, \(14\) is correct. \(12\) is only the value of \(3\times4\); the additional \(2\) must also be included. Exam tip: For an \(n\)th-term question, first substitute the given value of \(n\) correctly into the rule.

Count the terms in order: 3 is first, 6 second, 9 third, 12 fourth, and 15 fifth. Hence Reena has stated the position incorrectly. Exam tip: always begin numbering from the first listed term.

Given (a_n=12-n), substitute (n=5) to find the fifth term: (a_5=12-5=7). Therefore, the correct answer is 7. Getting 6 may result from incorrect subtraction or using the wrong term number. Exam tip: for an nth-term formula, substitute the given value of (n) carefully first.

To find the first term, substitute \(n=1\) in the given formula. Thus, \(a_1=5(1)-2=5-2=3\). Therefore, the correct answer is 3. The value 2 may result from an incorrect calculation of \(5-2\). Exam tip: for the \(k\)th term of a sequence, always put \(n=k\) in the formula.

Substituting \(n=1,2,3,4\) in the rule gives \(a_1=4(1)-1=3\), \(a_2=7\), \(a_3=11\), and \(a_4=15\). Therefore, the first four terms are \((3,7,11,15)\). In option B, the subtraction of 1 has been ignored. Exam tip: for the first term of a sequence, usually begin with \(n=1\), not \(n=0\).

For the first three terms, substitute \(n=1,2,3\). This gives \(a_1=1^2-1=0\), \(a_2=2^2-1=3\), and \(a_3=3^2-1=8\). Hence, the correct sequence is \((0,3,8)\). Option \((1,4,9)\) uses only \(n^2\) and does not subtract 1. Exam tip: to find initial terms from an nth-term rule, substitute 1, 2, and 3 in order and check each calculation.

For the first term, take \(n=1\). Thus, \(a_1=2^1=2\), \(a_2=2^2=4\), \(a_3=2^3=8\), and \(a_4=2^4=16\). Hence, the correct sequence is \((2, 4, 8, 16)\). Option A starts with \(2^0\), but the first term conventionally corresponds to \(n=1\). Exam tip: To list the first four terms, substitute \(n=1,2,3,4\) in the given rule.

This is an arithmetic sequence because 4 is added to each successive term. The terms are 8, 12, 16, 20, 24, 28; therefore, the 6th term is 28. Although 24 is a close distractor, it is the 5th term. Exam tip: write the terms in order and count their positions carefully.

In this sequence, each successive term is 6 less than the previous one: 90, 84, 78, 72, 66, 60. Therefore, the 6th term is 60. The number 66 is the 5th term, so it is a close but incorrect option. Exam tip: Count the first listed term as the 1st term.

Each term of this sequence is a multiple of 5, so its nth term is \(5n\). Putting \(5n=35\) gives \(n=7\); hence, 35 is the 7th term. The 6th term is \(30\), so the closest distractor is incorrect. Exam tip: To find a term’s position, equate the general term to the given value.

This is an arithmetic sequence with first term 3 and common difference 4. Its terms are 3, 7, 11, 15, 19; therefore, 19 is the fifth term. The fourth term is 15, so the fourth option is not correct. Exam tip: list the terms in order and count their positions.

Each successive term is 4 less than the preceding term: 48, 44, 40, 36, 32. Hence, 32 is the fifth term. The fourth term is 36, so the fourth-term option is incorrect. Exam tip: To find a term’s position, write the sequence with term numbers 1, 2, 3, and so on.

In the sequence (6, 12, 18, 24), the first term is 6, the second is 12, and the third is 18. Therefore, option A is correct. In option C, 18 is the second term, not the third. In exams, count terms from the left to identify their positions.

In the sequence (7, 12, 17, 22), the first term is 7 and the consecutive differences are 12 - 7 = 5, 17 - 12 = 5, and 22 - 17 = 5. Therefore, option B is correct. In option A, 4 is added each time, not 5. In exams, check both the first term and the difference between consecutive terms.

In the sequence \(64, 56, 48, 40\), the differences between consecutive terms are \(56-64=-8\), \(48-56=-8\), and \(40-48=-8\). Thus, 8 is subtracted each time. In option D, the terms decrease by 16, so it is not correct. Exam tip: Find the difference between consecutive terms to check a sequence rule quickly.

The terms of the sequence are 13, 17, 21, 25, \(\ldots\). Here, \(a_3\) denotes the third term, so \(a_3=21\). The number 25 is the fourth term, not the third. Exam tip: the subscript \(n\) in \(a_n\) indicates the position of the term.

The terms of the sequence are 70, 63, 56, and 49 in order. Therefore, the fourth term, \(a_4\), is \(49\). The number \(56\) is the third term, so it cannot be \(a_4\). Exam tip: To identify \(a_n\), count the terms from the left as 1, 2, 3, and so on.

In a sequence, \(a_n\) denotes the \(n\)th term. Hence, \(a_2\) is the second term. The given terms are 2, 11, 20, and 29, so \(a_2=11\). The number 20 is the third term, \(a_3\), not the second term. Exam tip: Count terms from the first term when reading sequence notation.

In a sequence, \(a_1\) denotes the first term. The first term of \(18,24,30,36,\ldots\) is 18, so 18 is correct. Here, 24 is the second term, \(a_2\), not \(a_1\). Exam tip: \(a_1\), \(a_2\), and \(a_3\) represent the first, second, and third terms respectively.

The differences between consecutive terms are 2, 3, 4, and 5. Since each difference increases by 1, the next difference is 6. Therefore, the next term is \(15+6=21\). Choosing 20 would give a difference of 5, but the required next difference is 6. Exam tip: For such sequences, list consecutive differences first and look for their pattern.