Sequences

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 \(n\).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 \(a_n=a_1+(n-1)d\).

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 \(n=1\).

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.

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.

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.

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.

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 \(n\).

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