Sequences

In this sequence, each term is half of the preceding term: 384, 192, 96, 48, 24, 12. Therefore, 24 comes immediately before 12. Option 36 is incorrect because half of 36 is 18, not 12. Exam tip: identify the common multiplication or division rule to find nearby terms in a sequence.

This is an arithmetic progression with first term \(a=17\) and common difference \(d=23-17=6\). Therefore, \(a_n=a+(n-1)d=17+(n-1)\times6=6n+11\). In \(6n+17\), substituting \(n=1\) gives 23, not the first term 17. Exam tip: identify the first term and common difference before applying \(a_n=a+(n-1)d\).

This is an arithmetic progression with first term \(a_1=108\) and common difference \(d=99-108=-9\). Thus, \(a_n=a_1+(n-1)d=108+(n-1)(-9)=117-9n\). In option B, putting \(n=1\) gives 99, so it does not match the first term. Exam tip: always test an nth-term rule by substituting \(n=1\).

Each term is 4 times the preceding term, so this is a geometric sequence. Its nth term is \(a_n=6\times4^{n-1}\). From \(6\times4^{n-1}=1536\), we get \(4^{n-1}=256=4^4\). Thus, \(n-1=4\) and \(n=5\). At position 4, the term is 384, not 1536. Exam tip: use \(a_n=a r^{n-1}\) to find the position of a term in a geometric sequence.

Given \(a_n=4n^2-3n+2\), \(a_5=4(5)^2-3(5)+2=100-15+2=87\) and \(a_2=4(2)^2-3(2)+2=16-6+2=12\). Hence, \(a_5-a_2=87-12=75\). However, 75 is not among the given options, so the original question or its options contain an error and no option is correct. In an exam, calculate each term separately before subtracting.

The rule is: next term = 2 × previous term + 1. Therefore, the term after 31 is \(2\times 31+1=62+1=63\). Option 62 is only twice 31; it misses the addition of 1. Exam tip: In a recursive sequence, apply the complete rule to the last given term.

This is an arithmetic sequence because 6 is added to each term. Here, the first term is \(a=8\), the common difference is \(d=6\), and \(n=16\). Thus, \(a_n=a+(n-1)d=8+(16-1)\times6=8+90=98\). Therefore, 98 is correct. Choosing 96 results from using 14 steps instead of the required 15 steps from the first term to the 16th term. Exam tip: for the \(n\)th term of an arithmetic sequence, use \(a+(n-1)d\).

This is an arithmetic sequence because each successive term is 12 less than the previous one. Here, \(a=160\), \(d=-12\), and \(n=11\). Thus, \(a_{11}=a+(11-1)d=160+10(-12)=40\). The value 48 is the 10th term, since it is obtained after subtracting 12 only 9 times. Exam tip: use \(n-1\) common differences to find the \(n\)th term.

To find the seventh term, substitute n=7 in the rule: (a_7=4(7)^2-3(7)=4×49-21=196-21=175). Hence, the correct answer is 175. The value 196 is only 4×49; subtracting 3×7 is also necessary. Exam tip: evaluate the power first, then multiply and subtract.

Given \(a_n=n^3-2n\), substitute \(n=5\): \(a_5=5^3-2(5)=125-10=115\). Therefore, 115 is correct. A value such as 120 can result from not subtracting \(2n\) correctly. Exam tip: evaluate powers first, then multiplication, and finally subtraction.

Each term in the sequence is 9 greater than the previous term: 9, 18, 27, 36, 45. Therefore, the term after 27 is \(27+9=36\). Choosing 34 or 38 would not maintain the common difference of 9. Exam tip: Find the difference between consecutive terms to identify the pattern.

Check the differences between the given terms: \(152-138=14\) and \(138-124=14\). Thus, each successive term is 14 less than the previous term. Therefore, \(180-14=166\), so the blank is 166. If 164 were chosen, the next difference would be 12, so it would not maintain the pattern. Exam tip: In a missing-term sequence, first check the differences between consecutive given terms.

The terms do not increase by a fixed amount, so we examine their consecutive differences. They are \(12-5=7\), \(22-12=10\), \(35-22=13\), and \(51-35=16\). These differences are 7, 10, 13, 16, and each is 3 greater than the previous one. This gives a clear rule for the next step.

The next difference is \(16+3=19\). Adding this to the fifth term gives \(51+19=70\). Therefore option B is correct. The sequence can also be understood as having second differences equal to 3, but calculating the first differences is enough here. The answer is not obtained by adding a constant number directly to the original terms.

The amounts subtracted are \(5, 10, 15, 20\). Each subtraction increases by \(5\), so the next subtraction is \(25\). Therefore, the next term is \(200-25=175\). \(170\) would result from subtracting \(30\), which does not follow the pattern. Exam tip: For next-term questions, first check the differences between consecutive terms.

This is a geometric sequence in which each term is 4 times the preceding term. After the given terms, the 5th term is 384 × 4 = 1536, and the 6th term is 1536 × 4 = 6144. Therefore, 6144 is correct. The closest distractor, 1536, is the 5th term, not the 6th. Exam tip: identify the common multiplier to find subsequent terms quickly.

This is a geometric sequence in which each term is one-third of the preceding term: \(729\div3=243\), \(243\div3=81\), and \(81\div3=27\). Therefore, the next term is \(27\div3=9\). Neither 6 nor 12 follows the same division rule. Exam tip: Compare consecutive terms to identify the pattern or common ratio.

The given terms are consecutive perfect squares: \(16=4^2, 25=5^2, 36=6^2, 49=7^2\), and \(64=8^2\). Hence, the \(n\)th term is \((n+3)^2\). Therefore, the \(8\)th term is \((8+3)^2=11^2=121\). Note that \(144=12^2\) is the next, or \(9\)th, term. Exam tip: For such sequences, take square roots of the terms to identify the underlying number pattern.

The terms are \(4^3, 5^3, 6^3, 7^3, \ldots\), respectively. Hence, the fifth term is \(8^3=512\), while the sixth term is \(9^3=729\). Therefore, 729 is correct. In exams, first identify the sequence of base numbers and then apply the exponent.

In option A, consecutive differences are \(7-3=11-7=15-11=4\), so it is an AP. Its ratios are not constant, hence it is not a GP. Exam tip: check differences before ratios.

By the given rule, each new term is the sum of the two terms immediately before it: 14 = 4 + 10, 24 = 10 + 14, and 38 = 14 + 24. Therefore, the next term is 24 + 38 = 62. An option such as 60 does not equal the sum of the previous two terms. Exam tip: for this type of sequence, add the last two given terms to find the next term.