Sequences

Correct answer: C. 77

Explanation: This is an arithmetic sequence because 5 is added to each successive term. Here, the first term is \(a=7\), the common difference is \(d=5\), and \(n=15\). Using \(a_n=a+(n-1)d\), \(a_{15}=7+(15-1)\times5=7+70=77\). Hence, 77 is correct. A common error is to use an incorrect count of gaps between the first and fifteenth terms. Exam tip: in the nth-term formula, use \((n-1)\), not \(n\).

Correct answer: B. 39

Explanation: This is an arithmetic sequence because each successive term decreases by 9. Here, the first term is \(a=120\) and the common difference is \(d=-9\). Thus, \(a_{10}=a+(10-1)d=120+9(-9)=39\). Choosing 30 would result from counting the number of differences incorrectly. Exam tip: for the \(n\)th term, use \(n-1\) common differences.

Correct answer: D. 96

Explanation: The general term is \(a_n=3n^2-2n\). Substituting \(n=6\), we get \(a_6=3(6)^2-2(6)=3\times36-12=108-12=96\). Hence, 96 is the correct option. An answer such as 94 may result from an error in squaring or multiplication. Exam tip: calculate \(n^2\) first, then perform multiplication and subtraction.

Correct answer: C. 68

Explanation: Given \(a_n=n^3+n\), substitute \(n=4\) to find the fourth term: \(a_4=4^3+4=64+4=68\). Therefore, 68 is correct. The value 64 is only \(4^3\); the additional \(+4\) must also be included. Exam tip: To find a particular term, substitute its index into the given formula before simplifying.

Correct answer: C. 25

Explanation: Check the differences between consecutive terms: \(11-4=7\) and \(18-11=7\). Thus, this is an arithmetic sequence in which 7 is added each time. Therefore, \(18+7=25\), and the check is \(25+7=32\). Hence, the missing term is 25. If 24 were chosen, the next difference would become 8, so it would not fit the pattern. Exam tip: For missing-term questions, first check the differences between consecutive terms.

Correct answer: C. 138

Explanation: This is a decreasing sequence in which each term is 12 less than the previous term: \(150-12=138\) and \(138-12=126\). Therefore, the missing value is 138. Choosing 136 would make the difference from 150 equal to 14, so it would not fit the common-difference pattern. Exam tip: In a missing-term sequence, compare consecutive known terms to identify the rule.

Correct answer: C. (61)

Explanation: The sequence does not have a constant first difference, so it is not an ordinary arithmetic progression. Instead, calculate the differences between consecutive terms. From 6 to 11 the difference is 5; from 11 to 19 it is 8; from 19 to 30 it is 11; and from 30 to 44 it is 14. These differences form the pattern 5, 8, 11, 14, increasing by 3 each time.

The next difference should therefore be \(14+3=17\). Adding it to the last known term gives \(44+17=61\). Hence option C is correct. A choice such as 58 would assume a smaller or constant increase and does not continue the observed difference pattern. The reliable method is to inspect consecutive differences before predicting the next term.

Correct answer: C. 155

Explanation: The differences between consecutive terms are \(200-197=3\), \(197-191=6\), \(191-182=9\), and \(182-170=12\). The numbers being subtracted increase by 3 each time, so the next subtraction is 15. Therefore, the next term is \(170-15=155\). Option 153 would require the next difference to be 17, which does not follow the given pattern. Exam tip: For such sequences, first list the differences between consecutive terms and look for their pattern.

Correct answer: B. 4

Explanation: Each term is one-fourth of the preceding term, so the previous term is divided by 4: \(1024\div4=256\), \(256\div4=64\), and \(64\div4=16\). Therefore, the next term is \(16\div4=4\). Getting 8 would require dividing 16 by 2, which does not follow the pattern. Exam tip: Check the ratio of consecutive terms to identify a geometric sequence rule.

Correct answer: B. 121

Explanation: The given terms are consecutive square numbers: \(9=3^2\), \(16=4^2\), \(25=5^2\), and so on. Therefore, the \(n\)th term is \((n+2)^2\). Hence, the \(9\)th term is \((9+2)^2=11^2=121\). Although \(144=12^2\), it is the next, or \(10\)th, term. Exam tip: In a sequence of squares, track the base numbers and match them with the term position.

Correct answer: B. 512

Explanation: The terms are \(3^3, 4^3, 5^3, 6^3, \ldots\). Hence, the \(n\)th term is \((n+2)^3\). Therefore, the (6)th term is \((6+2)^3=8^3=512\). Here, \(343=7^3\) is the fifth term, so it is a close but incorrect option. Exam tip: Rewrite the terms as cubes and identify the pattern in their bases.

Correct answer: B. 1, 4, 9, 16, \ldots

Explanation: For 1, 4, 9, 16, the first differences are 3, 5, 7 and the second differences are 2, 2. Since neither first differences nor ratios are constant, it is neither AP nor GP. Exam tip: check first differences first.

Correct answer: C. 51

Explanation: By the given rule, each new term is the sum of the two preceding terms: 11 = 2 + 9, 20 = 9 + 11, and 31 = 11 + 20. Therefore, the next term is 20 + 31 = 51. Option 52 is incorrect because it is not the sum of the previous two terms. Exam tip: In a sequence question, first verify the rule using the given terms before finding the next term.

Correct answer: C. 68

Explanation: The sequence follows the rule that each new term is the sum of the two preceding terms: 6+10=16, 10+16=26, and 16+26=42. Therefore, the next term is 26+42=68. Although 66 can be obtained by adding 24 to 42, it does not follow the sequence rule. Exam tip: In such sequences, first check consecutive differences; here, the differences are also equal to the preceding terms.

Correct answer: B. 5th term

Explanation: Each term in the sequence is 4 times the preceding term: 5, 20, 80, 320, 1280. Therefore, 1280 is the 5th term. The 4th term is 320, so it is a close but incorrect option. Exam tip: write the successive terms and count their positions carefully.

Correct answer: C. 10th term

Explanation: This is an arithmetic progression with first term 11 and common difference 8. Therefore, \(a_n=11+(n-1)\times8\). Putting \(a_n=83\), we get \(11+(n-1)\times8=83\), so \(n-1=9\) and \(n=10\). Hence, 83 is the 10th term. The 9th term is 75, so option B is not correct. Exam tip: To find the position of a term, first identify the first term and the common difference.

Correct answer: B. 8th term

Explanation: Each successive term is 9 less than the previous term. Hence, the nth term is \(140-9(n-1)\). Putting \(140-9(n-1)=77\) gives \(9(n-1)=63\), so \(n=8\). Therefore, 77 is the 8th term. The 9th term would be 68, so that option is incorrect. Exam tip: first identify the common difference, then form the nth-term expression to find a position.

Correct answer: A. \(a_n=9n-3\)

Explanation: The differences between consecutive terms are \(15-6=9\), \(24-15=9\), and \(33-24=9\), so this is an arithmetic progression. Its nth term is \(a_n=a_1+(n-1)d=6+(n-1)\times9=9n-3\). In option B, substituting \(n=1\) gives 15, so it does not match the first term. Exam tip: first find the common difference, then use \(a_n=a_1+(n-1)d\).

Correct answer: A. \(a_n=8n+5\)

Explanation: This is an arithmetic sequence because the difference between consecutive terms is \(8\). Here \(a_1=13\), so \(a_n=a_1+(n-1)d=13+(n-1)\times 8=8n+5\). Hence, option A is correct. In option B, putting \(n=1\) gives the first term as \(21\), so it is incorrect. Exam tip: always check a proposed nth-term rule by substituting \(n=1\).

Correct answer: A. (3, 10, 17, 24)

Explanation: For the first four terms, substitute n=1,2,3,4. This gives a_1=7(1)-4=3, a_2=7(2)-4=10, a_3=17, and a_4=24. Hence, the correct sequence is (3, 10, 17, 24). Option B ignores the subtraction of 4. Exam tip: for a sequence defined by a_n, usually begin with n=1 for the first term.

Correct answer: C. 75

Explanation: To find the fifth term, substitute \(n=5\) in the rule: \(a_5=2(5)^2+5(5)=2\times25+25=50+25=75\). Therefore, 75 is correct. A value such as 70 may result from calculating \(2\times25\) incorrectly. Exam tip: evaluate powers first, then multiplication, and finally addition.

Correct answer: A. \(a_n=5n-1\)

Explanation: For \(a_n=5n-1\), \(a_{n+1}-a_n=[5(n+1)-1]-(5n-1)=5\), a constant difference; hence it is an arithmetic progression. In \(n^2+1\), the differences change. Exam tip: compare consecutive terms’ differences.

Correct answer: B. (6, 24, 54, 96)

Explanation: Here, a_n=6n^2. Substituting n=1,2,3,4 gives 6(1)^2=6, 6(2)^2=24, 6(3)^2=54, and 6(4)^2=96. Therefore, the correct sequence is (6, 24, 54, 96). Option C gives only the values of n^2; it does not multiply them by 6. Exam tip: For a sequence rule, substitute the first few natural-number values of n to obtain its terms.

Correct answer: A. (3, 6, 11, 20)

Explanation: Here, \(a_n=2^n+n\), and the first term is obtained by taking \(n=1\). Thus, \(a_1=2^1+1=3\), \(a_2=2^2+2=6\), \(a_3=2^3+3=11\), and \(a_4=2^4+4=20\). Therefore, the correct sequence is \((3, 6, 11, 20)\). Option C is close, but its third term is incorrect because \(2^3+3=11\), not 12. Exam tip: evaluate the power first and then add \(n\).

Correct answer: C. (3)

Explanation: Both sequences have common difference (6), so the term difference always remains (3). The (10)th terms also differ by (3).