Sequences

Each term in the sequence is 3 greater than the previous term: 7−4=3, 10−7=3, and 13−10=3. Therefore, the next term is 13+3=16. Choosing 17 would require adding 4 to 13, which does not follow the pattern. Exam tip: Find the difference between consecutive terms to identify a sequence pattern.

Each consecutive term decreases by 5: from 60 to 55, 55 to 50, and 50 to 45. Therefore, the next term is \(45-5=40\). While 35 would be the term after that, 45 and 50 are already given terms. Exam tip: To find the next term, first identify the common difference between consecutive terms.

In this sequence, each term is 3 times the preceding term: \(3\to9\to27\to81\). Therefore, the next term is \(81\times3=243\). \(216\) is incorrect because it is not 3 times 81. Exam tip: first check whether consecutive terms follow a multiplication or division pattern.

The difference between consecutive terms is 4: 5−1=4, 9−5=4, and 13−9=4. Therefore, the term after 13 is 13+4=17. Choosing 16 would give a difference of 3, so it is incorrect. Exam tip: check the difference between consecutive terms before finding the next term.

In this sequence, each term is 6 more than the previous term: 6, 12, 18, 24, 30. Therefore, the term after 18 is 24. Choosing 22 or 20 would not maintain the common difference of 6. Exam tip: For a missing-term sequence, first check the difference between consecutive known terms.

In this sequence, each next term is 10 less than the previous term: 100, 90, 80, 70, 60. Therefore, the term after 90 and before 70 is 80. Choosing 75 would not keep the difference between consecutive terms constant. Exam tip: In a missing-term sequence, compare consecutive terms to identify the rule.

The rule for the terms is \(a_n=4n\). To find the fifth term, substitute \(n=5\): \(a_5=4\times 5=20\). Hence, 20 is correct. The value 16 comes from \(4\times4\), which is the fourth term \(a_4\), not the fifth term. Exam tip: the subscript in \(a_n\) shows the term position, so substitute that number for \(n\).

Given \(a_n=n^2+1\), substitute \(n=3\): \(a_3=3^2+1=9+1=10\). Therefore, 10 is correct. Option 9 is only the value of \(3^2\); it misses the added \(+1\). Exam tip: To find a term of a sequence, first substitute the term number for \(n\), then simplify carefully.

To find the common difference, subtract a term from the term immediately after it: \(36-42=-6\). Similarly, \(30-36=-6\), so the common difference between consecutive terms is \(-6\). Although the terms decrease by 6, the difference itself must be negative. Exam tip: always calculate common difference as second term minus first term.

To get 6 from 2, 18 from 6, and 54 from 18, the preceding term is multiplied by 3 each time: \(2\times3=6\), \(6\times3=18\), and \(18\times3=54\). Therefore, multiplying by 3 is correct. Adding 6 works only for the first pair, since \(6+6=12\), not 18. Exam tip: check a sequence rule using at least two consecutive pairs of terms.

In the sequence, 80 becomes 40, 40 becomes 20, and 20 becomes 10 by dividing the preceding term by 2 each time. Therefore, the correct rule is division by 2. Subtracting 10 would give 70 after 80, while dividing by 4 would give 20 after 80. Exam tip: Compare consecutive terms using their ratio to quickly identify a multiplication or division pattern.

The terms are squares of consecutive natural numbers: \(1^2, 2^2, 3^2, 4^2, 5^2\). Therefore, the next term is \(6^2=36\). Although 49 is also a square number, it is \(7^2\) and comes after 36. Exam tip: check consecutive differences in such sequences; here they are \(3,5,7,9\), so the next difference is 11.

The terms are consecutive cubes of natural numbers: \(1=1^3\), \(8=2^3\), \(27=3^3\), \(64=4^3\), and \(125=5^3\). Therefore, the next term is \(6^3=216\). Note that \(225\) is not a perfect cube. Exam tip: Rewrite sequence terms as \(1^3, 2^3, 3^3\), and so on, to identify the pattern quickly.

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 \(16+6=22\). Choosing 21 would give a difference of only 5, whereas the differences should follow 2, 3, 4, 5, 6. Exam tip: For such sequences, first write the consecutive differences and identify their pattern.

The successive terms are obtained by subtracting 1, 2, 3, and 4: 30 to 29, 29 to 27, 27 to 24, and 24 to 20. Therefore, the next subtraction is 5: \(20-5=15\). Hence, 15 is the correct answer. Option 16 would result from subtracting 4 again, but the pattern of subtractions is 1, 2, 3, 4, 5. Exam tip: For such sequences, write the differences between consecutive terms and look for their pattern.

In this sequence, each term is the sum of the two immediately preceding terms: \(1+1=2\), \(1+2=3\), \(2+3=5\), and \(3+5=8\). Therefore, the next term is \(5+8=13\). Although 12 may seem like a simple continuation after 8, it does not follow the sequence rule. Exam tip: For such sequences, check whether adding consecutive previous terms produces the next term.

From the third term onward, each term is the sum of the two immediately preceding terms: \(9=4+5\), \(14=5+9\), and \(23=9+14\). Therefore, the next term is \(14+23=37\). Option 36 would require adding 14 and 22, which does not follow the pattern. Exam tip: For sequence questions, first check successive differences and then check sums of adjacent terms.

In the sequence (32, 28, 24, 20), the differences between consecutive terms are 28 − 32 = −4, 24 − 28 = −4, and 20 − 24 = −4. Hence, 4 is subtracted each time. In option B, 6 is subtracted each time, so it is not correct. Exam tip: Find the difference between consecutive terms to check the rule of a sequence.

(6,12,18,24) are all multiples of (6). In a multiples sequence, the first term and difference both help.

In option C, the terms are \(1^3=1\), \(2^3=8\), \(3^3=27\), and \(4^3=64\). Therefore, it is a sequence of cube numbers. Option A is a sequence of square numbers, since its terms are \(1^2, 2^2, 3^2, 4^2\). Exam tip: find a cube by multiplying a number by itself three times; for example, \(3^3=3\times3\times3=27\).

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 \(n=0\).