nth term

Correct answer: B. aₙ = 5n

Explanation: The key concept is recognizing a sequence whose terms are consecutive multiples of 5. The first term is 5, the second is 5 × 2, the third is 5 × 3, and the fourth is 5 × 4. Therefore the term in position n is 5 × n, or aₙ = 5n. Substitution verifies the rule: n = 1 gives 5, n = 2 gives 10, n = 3 gives 15, and n = 4 gives 20. Hence option B is correct. Option A gives 6 as the first term, option C gives 10 as the first term and grows twice as fast, and option D produces square numbers rather than the displayed multiples of 5. The rule must work for every position, not just one term.

Correct answer: B. \(15\)

Explanation: For the eighth term, substitute \(n=8\): \(a_8=2(8)-1=16-1=15\). Therefore, the correct answer is \(15\). \(16\) would result if the \(-1\) in the formula were omitted. Exam tip: in an \(n\)th-term question, substitute the given value of \(n\) carefully into the formula.

Correct answer: C. 28

Explanation: Direct answer: the tenth term is 28, so C is correct. The sequence increases by 3 each time: 4 − 1 = 3, 7 − 4 = 3, and 10 − 7 = 3. It is therefore an arithmetic sequence with first term a₁ = 1 and common difference d = 3. The nth-term formula is aₙ = a₁ + (n − 1)d. For n = 10, a₁₀ = 1 + (10 − 1)3 = 1 + 27 = 28. Equivalently, the rule simplifies to aₙ = 3n − 2, and 3(10) − 2 = 28. Option A, 25, is the ninth term because 1 + 8×3 = 25. B does not fit the repeated difference of 3. D, 30, comes from multiplying the position by 3 and forgetting that the sequence starts at 1 rather than 3. Memory cue: from the first term to the tenth there are nine jumps, not ten jumps.

Correct answer: C. 42

Explanation: For the sixth term, substitute \(n=6\) in the formula: \(a_6=7\times6=42\). Therefore, 42 is the correct answer. The value 49 would be obtained for \(n=7\), so it is the seventh term. Exam tip: In an nth-term question, carefully substitute the required term number into the formula.

Correct answer: A. aₙ = 5n + 1

Explanation: The sequence is arithmetic because each term increases by the constant difference 5. With first term a₁ = 6 and common difference d = 5, the nth-term formula is aₙ = a₁ + (n − 1)d. Substitution gives aₙ = 6 + 5(n − 1) = 6 + 5n − 5 = 5n + 1. Therefore option A is correct. Verification is important: n = 1 gives 5(1) + 1 = 6, n = 2 gives 11, and n = 4 gives 21. Option B gives 5 when n = 1, option C gives 4, and option D has a common difference of only 1. These checks show why the selected expression is the only one matching all displayed terms.

Correct answer: C. \(25\)

Explanation: For the seventh term, substitute \(n=7\): \(a_7=3\times 7+4=21+4=25\). Therefore, the correct answer is \(25\). \(23\) may result from an error in multiplication or addition. Exam tip: after substituting the term number, multiply first and then add.

Correct answer: C. (a_n=10n)

Explanation: This is a sequence of multiples of (10), so (a_n=10n). In exams, identify a sequence of multiples quickly.

Correct answer: C. 17

Explanation: For the ninth term, substitute \(n=9\) in the rule: \(a_9=9+8=17\). Therefore, the correct answer is 17. The value 16 would result from using \(n=8\), which gives the eighth term. Exam tip: always substitute the term number asked for in place of \(n\).

Correct answer: A. (a_n=5n+4)

Explanation: The difference is (5), and the rule must give (9) at (n=1), so (a_n=5n+4). In exams, check the rule on the first two terms.

Correct answer: B. 28

Explanation: For the fifth term, substitute \(n=5\) in the formula: \(a_5=6\times5-2=30-2=28\). Therefore, 28 is correct. The close distractor 30 results from forgetting to subtract 2. Exam tip: substitute the value of \(n\) first, then follow the correct order of operations.

Correct answer: B. (a_n=2^n)

Explanation: Direct answer: Option B, \\(a_n=2^n\\), is correct. This is a geometric sequence: each term is obtained by multiplying the previous term by 2. Check the positions carefully: first term is 2, and \\(2^1=2\\); second is 4, and \\(2^2=4\\); third is 8, and \\(2^3=8\\); fourth is 16, and \\(2^4=16\\). Thus the nth term is \\(2^n\\). Option A, \\(2n\\), gives 2, 4, 6, 8, so it stops matching at the third term. Option B matches every displayed term and is correct. Option C, \\(n^2\\), gives 1, 4, 9, 16, so it also fails. Option D, \\(2^{n-1}\\), gives first term 1 instead of 2; it would be correct for a sequence beginning with 1. Memory cue: match the first term with the exponent’s starting value.

Correct answer: B. 12

Explanation: To find the fourth term, substitute n=4: \(a_4=20-2(4)=20-8=12\). Therefore, the correct answer is 12. The value 14 is obtained for n=3, so it is the third term. Exam tip: carefully substitute the required term number for n in the nth-term formula.

Correct answer: A. 25

Explanation: This is an arithmetic progression with first term 7 and common difference 2. Therefore, the eleventh term is \(a_{11}=7+(11-1)\times2=27\). Hence, 27 is the correct option. In exams, remember to use \(n-1\) in the formula \(a_n=a+(n-1)d\).

Correct answer: C. 81

Explanation: For the fourth term, substitute n=4: \(a_4=3^4=3\times3\times3\times3=81\). Therefore, the correct answer is 81. The value 27 equals \(3^3\), so it is the third term. Exam tip: write the required term number in place of n before evaluating the expression.

Correct answer: B. aₙ = 8n

Explanation: The sequence consists of successive multiples of 8: 8 × 1, 8 × 2, 8 × 3, and 8 × 4. Thus the term in position n is 8 × n, so the general rule is aₙ = 8n. Option B is therefore correct. The rule can be checked directly: at n = 1 it gives 8, at n = 2 it gives 16, at n = 3 it gives 24, and at n = 4 it gives 32. Option A gives only half of each displayed term, option C does not preserve the constant difference of 8, and option D gives 16 as the first term and doubles the required values. A valid nth-term expression must reproduce every listed position consistently.

Correct answer: B. Each successive term is 3 less than the preceding term

Explanation: In \(a_n=7-3n\), the coefficient of \(n\) is \(-3\). Thus, when \(n\) increases by 1, the term decreases by 3; the common difference is \(-3\). Option A describes an increase. Exam tip: for \(pn+q\), the common difference is \(p\).

Correct answer: A. aₙ = 3n + 9

Explanation: Direct answer: aₙ = 3n + 9, so A is correct. Subtract neighboring terms: 15 − 12 = 3, 18 − 15 = 3, and 21 − 18 = 3. The constant difference shows that the sequence is arithmetic. Use aₙ = a₁ + (n − 1)d. With a₁ = 12 and d = 3, aₙ = 12 + (n − 1)3 = 12 + 3n − 3 = 3n + 9. Testing confirms it: at n = 1, the rule gives 12; at n = 2, it gives 15; and at n = 3, it gives 18. B gives 12 at n = 1 but then grows by 12, so it cannot match. C gives 15 for the first term, not 12. D gives a first term of 12 but has common difference 15, not 3. A useful check is to verify both the first term and the difference; matching only one term is not enough.

Correct answer: C. 26

Explanation: For the fifth term, substitute n=5: (a_5=5^2+1=25+1=26). Therefore, 26 is the correct answer. 25 is only the square of 5; the +1 from the formula must still be added. Exam tip: To find an nth term, first substitute the term number for n and then simplify.

Correct answer: B. (77)

Explanation: This is a sequence of multiples of (11), so (a_7=11(7)=77). In exams, identify multiples to solve quickly.

Correct answer: B. 23

Explanation: For the third term, substitute \(n=3\): \(a_3=9\times 3-4=27-4=23\). Therefore, the correct answer is 23. The value 21 may result from an error in subtraction. Exam tip: first substitute the term number into the \(n\)th-term formula, then simplify step by step.

Correct answer: A. (a_n=4n+9)

Explanation: This is an arithmetic sequence because every term increases by the same amount. The common difference is 4: 17−13=4, 21−17=4, and 25−21=4. For an arithmetic sequence, the nth term is found by starting with the first term and adding the common difference \\(n-1\\) times. Thus, \\(a_n=a_1+(n-1)d\\), where \\(a_1=13\\) and \\(d=4\\).

Substitution gives \\(a_n=13+(n-1)4=13+4n-4=4n+9\\). Therefore, option A is correct. A quick check confirms it: for \\(n=1\\), the formula gives \\(4(1)+9=13\\); for \\(n=2\\), it gives 17; and for \\(n=4\\), it gives 25. The other expressions do not produce the given first terms in the required order.

Correct answer: C. 4

Explanation: For the eighth term, substitute \(n=8\). Thus, \(a_8=\frac{8}{2}=4\). Therefore, the correct answer is 4. Option 2 would result from using \(n=4\), which represents the fourth term. Exam tip: substitute the required term number carefully in an nth-term formula.

Correct answer: B. 9

Explanation: For the second term, substitute n=2: \(a_2=15-3(2)=15-6=9\). Therefore, the correct answer is 9. The value 12 is obtained when \(n=1\), so it is the first term, not the second. Exam tip: In an nth-term formula, substitute the required term number carefully for \(n\).

Correct answer: B. (49)

Explanation: This starts from the second multiple of (7), and the sixth term is (49). In exams, you can also check by counting terms.

Correct answer: A. 1

Explanation: To find the first term, substitute \(n=1\) in the formula: \(a_1=4(1)-3=1\). Therefore, 1 is the correct option. Option 3 does not result from substituting \(n=1\); it may be chosen by confusing it with the constant part of the expression. Exam tip: For the first term of any sequence, put \(n=1\).