Sequences

The differences between consecutive terms are 3, 5, 7, and 9, which are consecutive odd numbers. Therefore, the next difference is 11. Hence, the next term is \(26+11=37\). Choosing 35 would give a difference of only 9 and would break the pattern of increasing odd differences. Exam tip: For such sequences, first write the consecutive differences to identify the pattern.

The differences between consecutive terms are \(7-5=2\), \(10-7=3\), \(14-10=4\), and \(19-14=5\). Since the difference increases by 1 each time, the next difference is \(6\). Therefore, the next term is \(19+6=25\). Option 24 would result if the next difference remained 5, which does not follow the pattern. Exam tip: Write the consecutive differences first to identify a sequence pattern.

In this sequence, each term is obtained by dividing the previous term by 2: 120 to 60, 60 to 30, and 30 to 15. Therefore, the next term is \(15 \div 2 = 7.5\). Option 8 is not half of 15. Exam tip: Check for a common multiplication or division rule between consecutive terms.

To find the third term, substitute \(n=3\) in the rule: \(a_3=6\times3+1=18+1=19\). Hence, 19 is the correct answer. The value 18 is only \(6\times3\), so it misses the \(+1\) in the rule. Exam tip: substitute the given value of \(n\) first, then carry out all operations carefully.

Given \(a_n=3n^2\), substitute \(n=2\): \(a_2=3\times2^2=3\times4=12\). Hence, option A is correct. Option B results from multiplying \(3\times2\) without squaring 2. Exam tip: while finding a term of a sequence, substitute the given value of \(n\) first and then evaluate the exponent.

To find the first three terms, substitute n=1, 2, and 3. This gives a_1=15-2(1)=13, a_2=15-2(2)=11, and a_3=15-2(3)=9. Therefore, the sequence begins with (13, 11, 9). The option (15, 13, 11) would result if indexing started at n=0. Exam tip: Unless stated otherwise, use n=1 to find the first term from a_n.

Each term in the sequence is twice the preceding term: 10, 20, 40, 80. Therefore, the next term is \(80\times 2=160\). The value 120 would result from adding 40 to 80, but the pattern is multiplication by 2, not addition. Exam tip: check the ratio of consecutive terms; here \(20/10=40/20=80/40=2\).

Write the sequence in pairs: (2, 4), (3, 6), (4, 8). In each pair, the second term is twice the first term, while the first terms 2, 3, 4, ... increase by 1. Therefore, the term after 8 is 5. Option 10 is not the next term; it would come after 5. Exam tip: In such sequences, examine the odd-position and even-position terms separately to identify the pattern.

The differences between consecutive terms are \(2, 4, 2, 4\). The pattern alternates between \(+2\) and \(+4\), so add \(+2\) after 13: \(13+2=15\). Therefore, 15 is the correct answer. Option 17 would result from adding 4 next, but the next difference in the pattern is 2. Exam tip: For next-term questions, first write the differences between consecutive terms to identify the pattern.

Each term in this sequence increases by 4, so the first term is 3 and the common difference is 4. Thus, \(a_{10}=3+(10-1)\times4=3+36=39\). The value 37 is obtained after adding 4 only eight times, so it is the 9th term. Exam tip: for the \(n\)th term, add the common difference \(n-1\) times to the first term.

This is an arithmetic progression with first term \(a=42\) and common difference \(d=-5\). The \(n\)th term is \(a_n=a+(n-1)d\). Therefore, \(a_8=42+(8-1)(-5)=42-35=7\). Hence, 7 is correct. Getting 12 usually results from counting one fewer decrease than required. Exam tip: for the \(n\)th term, count \(n-1\) common differences.

Given \(a_n=3n+4\), substitute \(n=7\) to find the seventh term: \(a_7=3\times7+4=21+4=25\). Hence, 25 is correct. The value 24 would result from using \(3\times7+3\), which does not match the given rule. Exam tip: For an \(a_n\) question, first substitute the required term number for \(n\), then simplify.

To find the fifth term, substitute n=5 in the formula: a_5=2(5)^2-1=2×25-1=49. Therefore, 49 is the correct answer. 51 may result from an error in the final subtraction. Exam tip: evaluate the power first, then multiply and subtract.

Check the difference between consecutive terms: 9 - 5 = 4 and 13 - 9 = 4. Thus, 4 is added to each term. Therefore, the term after 13 is 13 + 4 = 17, and 17 + 4 = 21 confirms the pattern. Option 16 is not correct because it would give 20 as the next term. Exam tip: In missing-term questions, first check the difference between consecutive terms.

Check the differences between the known terms: from 68 to 62 and from 62 to 56, the decrease is 6 each time. Thus, the rule is to subtract 6 for every next term. Therefore, the term after 80 is 80 - 6 = 74, and 74 - 6 = 68 also confirms the pattern. Choosing 72 would make the next difference 4, so it is incorrect. Exam tip: In missing-term sequences, first check the differences between consecutive given terms.

The differences between consecutive terms are \(6-2=4\), \(12-6=6\), \(20-12=8\), and \(30-20=10\). These differences increase by 2 each time, so the next difference is \(12\). Therefore, the next term is \(30+12=42\). Choosing \(40\) would repeat a difference of 10 instead of continuing the pattern. Exam tip: For such sequences, first list the consecutive differences and look for their pattern.

Look at the differences between consecutive terms: 2, 4, 6, and 8 are subtracted from 100. The subtracted even number increases by 2 each time, so the next subtraction is 10. Thus, 80 - 10 = 70. Option 72 would result from subtracting 8 again, but the subtraction pattern is 2, 4, 6, 8, 10. Exam tip: For such sequences, write the consecutive differences first and identify their pattern.

In this sequence, each term is twice the preceding term: 3, 6, 12, 24, 48, 96, 192. Therefore, the 7th term is 192. The number 96 is the 6th term, so it is a close but incorrect option. Exam tip: Count term positions starting with the first term as 1.

In this sequence, each term is obtained by dividing the preceding term by 3: \(243\div3=81\), \(81\div3=27\), and \(27\div3=9\). Therefore, the next term is \(9\div3=3\). Although 6 is obtained by subtracting 3 from 9, it does not follow the division rule of the sequence. Exam tip: First check for a multiplication or division relationship between consecutive terms.

This is a sequence of perfect squares: \(1^2, 2^2, 3^2, 4^2, 5^2, \ldots\). Hence, its \(n\)th term is \(n^2\). Therefore, the 9th term is \(9^2=81\). Note that 64 is \(8^2\), so it is the 8th term. Exam tip: for a sequence of perfect squares, square the term number to find the corresponding term.