nth term

Use the arithmetic-progression formula a_n = a + (n - 1)d. Here a = 18 and d = 25 - 18 = 7. Thus a_n = 18 + 7(n - 1) = 7n + 11. We are told that a_n = 102, so solve 7n + 11 = 102. Subtracting 11 gives 7n = 91, and dividing by 7 gives n = 13. Therefore 102 is the thirteenth term, making option C correct. Verification is immediate: 18 + (13 - 1)7 = 18 + 84 = 102. The nearby answers 11, 12, and 14 arise from an arithmetic error or from using n rather than n - 1 in the formula.

The governing relation is a_n = a_1 + (n - 1)d. For n = 7, the seventh term is a_7 = a_1 + 6d. Substitute the known values: 50 = a_1 + 6(9) = a_1 + 54. Therefore a_1 = 50 - 54 = -4. Option A is correct. The result can also be understood by moving backward six steps from the seventh term, subtracting 9 each time: 50, 41, 32, 23, 14, 5, -4. Options B, C, and D do not account correctly for all six common differences between the first and seventh terms; using only five differences, or adding instead of subtracting, leads to an incorrect answer.

For an arithmetic progression, use a_n = a + (n - 1)d. The first term is a = 19 and the common difference is d = 25 - 19 = 6. For n = 24, a_24 = 19 + (24 - 1)6 = 19 + 23(6). Since 23 multiplied by 6 is 138, the result is 19 + 138 = 157. Therefore option B is correct. The expression uses 23 differences, not 24, because the first term is already counted as the starting value. Option A is too small, while C and D result from adding one or two extra common differences or from an arithmetic mistake.

For the fifth term, substitute \(n=5\). Then \(a_5=2(5)+3=10+3=13\). Therefore, 13 is the correct answer. A value such as 15 may result from an error in multiplication or addition. Exam tip: in an \(n\)th-term question, first substitute the given term number carefully for \(n\).

For the fourth term, substitute n=4. Then a_4=5×4−1=20−1=19, so 19 is correct. The value 21 may result from adding 1 instead of subtracting it. Exam tip: after substituting n, perform multiplication before addition or subtraction.

This is a sequence of multiples of (3), so (a_n=3n). In exams, match the rule with the first few terms.

The governing concept is the nth-term formula for an arithmetic sequence. Consecutive terms increase by a constant difference of 3, so the sequence is arithmetic with first term a₁ = 4 and common difference d = 3. Its nth term is aₙ = a₁ + (n − 1)d = 4 + 3(n − 1) = 4 + 3n − 3 = 3n + 1. Thus option B is correct. A quick check confirms it: for n = 1, 3(1) + 1 = 4; for n = 2, it gives 7; and for n = 4, it gives 13. Option A gives 2 for n = 1, option C gives 4n, and option D has the wrong growth rate.

For the sixth term, substitute \(n=6\): \(a_6=10-6=4\). Therefore, the correct answer is 4. Option 5 would result from using \(n=5\), which gives the fifth term. Exam tip: In an nth-term formula, first substitute the correct term number.

For the seventh term, substitute n=7. Thus, a_7=7^2=49, so the correct answer is 49. The value 64 equals 8^2, so it would be the eighth term. Exam tip: substitute the given term number into the nth-term formula before calculating.

The first term is (2) and the difference is (3), so (a_n=2+(n-1)3=3n-1). In exams, do not forget to simplify the formula.

For the third term, substitute n=3: \(a_3=4\times3+2=12+2=14\). Therefore, 14 is correct. The value 12 is only \(4\times3\); the +2 must also be added. Exam tip: In an nth-term question, first substitute the required value of n correctly into the formula.

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.

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.

The governing concept is the nth term of an arithmetic sequence. The first term is a₁ = 1 and the common difference is d = 3. Therefore aₙ = a₁ + (n − 1)d = 1 + 3(n − 1) = 1 + 3n − 3 = 3n − 2. For the tenth term, put n = 10: a₁₀ = 3(10) − 2 = 30 − 2 = 28. Hence option C is correct. A direct continuation also gives the same value: the ninth term is 25 and adding 3 gives the tenth term 28. Option A is the ninth term, option B does not follow the constant difference, and option D forgets the subtraction of 2 in the general rule.

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.

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.

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.

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

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 \(n\).

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.