nth term

The governing idea is finding an explicit rule from a quadratic pattern. The successive differences are 2, 4, 6, and 8, whose second difference is constant at 2, so the nth term has the form n^2 + bn + c. Testing the displayed terms gives a_n = n^2 - n + 3: for n = 1, 2, 3, 4, 5 it gives 3, 5, 9, 15, 23 respectively. Substituting n = 12, a_12 = 12^2 - 12 + 3 = 144 - 12 + 3 = 135. Hence option B is correct. The nearby alternatives 132, 138, and 141 can arise from using an incorrect linear continuation or making an arithmetic error while substituting 12.

The governing concept is evaluation of an explicit nth-term formula, including both the cube and the square of the index. For n = 6, substitute carefully: a₆ = 6³ + 6² = 216 + 36 = 252. For n = 4, a₄ = 4³ + 4² = 64 + 16 = 80. Hence a₆ − a₄ = 252 − 80 = 172, making option C correct. The two terms should be calculated separately before subtraction. Omitting the square or cube, using an incorrect power, or making an arithmetic error in the final subtraction can produce distractors such as 162, 168, or 176. Since the formula contains a sum of two powers, both components must be included for each index.

The governing concept is finding the position of a term from an explicit nth-term rule. The required term has value 75, so substitute aₙ = 75 in the formula: 4n + 7 = 75. Subtracting 7 from both sides gives 4n = 68. Dividing by 4 gives n = 17. Therefore, 75 is the seventeenth term and option C is correct. Verification using the original formula gives a₁₇ = 4(17) + 7 = 68 + 7 = 75. The distractors are nearby values but fail the rule: n = 15 gives 67, n = 16 gives 71, and n = 18 gives 79. Thus only n = 17 produces exactly the required term.

This question applies an explicit nth-term rule in reverse: instead of finding a term from n, we find n from a given term value. Set 5n+3=88 because the required term equals 88. Subtract 3 from both sides to obtain 5n=85, then divide by 5 to get n=17. Thus 88 is the seventeenth term, so option C is correct. Substitution verifies the answer: a₁₇=5(17)+3=85+3=88. The alternatives do not satisfy the equation: n=15 gives 78, n=16 gives 83, and n=18 gives 93. Maintaining the equality during each inverse operation is the essential algebraic reasoning.

The governing concept is recognizing a pattern and expressing each term directly in terms of its position n. The displayed terms can be rewritten as 1 = 1³, 8 = 2³, 27 = 3³, and 64 = 4³. Thus the first term is the cube of 1, the second is the cube of 2, and so on. Therefore, the nth term is aₙ = n³, making option C correct. Option A represents square numbers and would produce 1, 4, 9, 16. Option B represents powers of 2 and would produce 2, 4, 8, 16 when n starts at 1. Option D uses (n + 1)³, whose first term would be 8 rather than 1. Checking the first and second terms quickly distinguishes the correct rule from these distractors.

This is an arithmetic sequence because the difference between consecutive terms is constant: 13 − 7 = 6, 19 − 13 = 6, and 25 − 19 = 6. For an arithmetic sequence, the nth-term formula is aₙ = a₁ + (n − 1)d. Here a₁ = 7 and d = 6, so aₙ = 7 + 6(n − 1) = 6n + 1. Substituting n = 25 gives a₂₅ = 6(25) + 1 = 150 + 1 = 151. Therefore option C is correct. Option B is one less than the correct value, while option D is one greater. Option A results from using an incorrect difference or index adjustment. There are 24 equal jumps from the first term to the 25th term, which explains the factor n − 1.

The governing skill is evaluating an explicit fractional rule at a specified index. To find a_4, replace every n in both the numerator and denominator by 4: a_4=(2(4)+3)/(4+1)=(8+3)/5=11/5. Therefore option C is correct. A common error is to substitute 4 only in the numerator, only in the denominator, or to add the numerator and denominator instead of forming their quotient. Option A would correspond to an incorrect numerator, option B simplifies to 2 and does not equal the calculated fraction, and option D uses 12 rather than 11 in the numerator. Keeping the fraction unsimplified also makes the substitution transparent.

The governing concept is identifying an explicit nth-term rule by recognizing perfect cubes and matching their bases with the term positions. The sequence can be written as 27 = 3³, 64 = 4³, 125 = 5³, and 216 = 6³. At position n = 1, the cube base is 3; at n = 2, it is 4. Thus the base is always n + 2, giving aₙ = (n + 2)³. Checking confirms that n = 1 gives 3³ = 27 and n = 4 gives 6³ = 216. Option A starts with 1³ = 1, and option B starts with 2³ = 8, so neither matches. Option D gives 1³ + 26 = 27 initially but does not reproduce the later cube values. Therefore option C is correct.

The governing concept is deriving a general term from a pattern of perfect squares. Rewrite the terms as 4 = 2², 36 = 6², 100 = 10², and 196 = 14². The square bases form an arithmetic sequence: 2, 6, 10, 14, with common difference 4. The nth base is therefore 2 + (n − 1)4 = 4n − 2. Squaring it gives aₙ = (4n − 2)², so option B is correct. Substitution verifies the rule: n = 1 gives 2² = 4, n = 2 gives 6² = 36, n = 3 gives 10² = 100, and n = 4 gives 14² = 196. Option A produces 4, 16, 36, 64; option C starts at 16; and option D does not generate the sequence.

The governing concept is the nth-term formula of an arithmetic sequence. Consecutive terms increase by 7, so the sequence has first term a₁ = 12 and common difference d = 7. Its explicit rule is aₙ = a₁ + (n − 1)d = 12 + 7(n − 1) = 7n + 5. Substituting n = 30 gives a₃₀ = 7(30) + 5 = 210 + 5 = 215. Therefore option B is correct. Option A is 7 less than the correct result and would come from using 29 without the required first-term adjustment; C and D are obtained by adding too much. The explicit formula avoids listing all thirty terms and is the efficient method.

The governing concept is verifying an explicit general-term formula by substituting the first several positive integer values of n. For option A, n = 1 gives 1(2 + 3)/2 = 5/2. For n = 2, it gives 2(4 + 3)/2 = 7. For n = 3, it gives 3(6 + 3)/2 = 27/2. For n = 4, it gives 4(8 + 3)/2 = 22. Thus option A reproduces every displayed term exactly and is the correct rule. Option B gives 2 when n = 1, option C gives 3, and option D gives 2; each fails at the first term itself. Testing several indices is especially important when fractions are present, because a formula can appear similar while having an incorrect coefficient or constant.

The governing concept is verifying a proposed explicit general term by substituting successive positive integers. For option A, n = 1 gives 5 − 2 = 3; n = 2 gives 25 − 4 = 21; n = 3 gives 125 − 6 = 119; and n = 4 gives 625 − 8 = 617. Every displayed term is reproduced, so a_n = 5^n − 2n is correct. Option B grows only linearly, option C gives 3, 5, 9, 17, and option D gives negative values initially or does not match the sequence. Checking several indices prevents accepting a formula that fits only one term.

The governing concept is finding and verifying an nth-term formula, including careful handling of fractional terms. Substitute successive values into option A. When n = 1, a₁ = (2 + 5)/3 = 7/3. When n = 2, a₂ = (8 + 10)/3 = 18/3 = 6. When n = 3, a₃ = (18 + 15)/3 = 33/3 = 11. When n = 4, a₄ = (32 + 20)/3 = 52/3. Thus option A reproduces the complete given sequence. Option B gives the triangular numbers 1, 3, 6, 10; option C gives 3, 5, 7, 9; and option D gives 2, 13/2, 14, 25/2. Since these do not match the displayed values, A is the only correct answer.

Given \(a_n=3n^2-2n+5\), substitute \(n=8\): \(a_8=3(8)^2-2(8)+5=3\times64-16+5=181\). Therefore, 181 is correct. A value such as 189 can result from an error while evaluating \(8^2\) or performing the subtraction. Exam tip: evaluate the power first, then multiply, subtract, and add.

The consecutive differences are \(5,7,9,11\), increasing by 2, so the rule is quadratic. Using \(a_n=n^2+2n-1\) gives \(a_1=2\), \(a_2=7\), \(a_3=14\), and \(a_4=23\). Although \(n^2+1\) gives the first term as 2, it gives the second term as 5, not 7. Exam tip: test an nth-term formula with at least the first two or three terms.

The kth term is given as 47. Substitute k for n in the formula: \(5k-3=47\). Thus, \(5k=50\), so \(k=10\). If 9 is used, the term is \(5\times9-3=42\), not 47. Exam tip: To find a term number, equate the nth-term formula to the given term value and solve for n.

For an arithmetic sequence, \(a_n=a_1+(n-1)d\). The difference between the given terms is \(a_9-a_4=43-18=25\), while the difference between their positions is \(9-4=5\). Hence, \(d=25/5=5\). Using \(a_4=a_1+3d\), we get \(18=a_1+15\), so \(a_1=3\). Therefore, \(a_n=3+(n-1)5=5n-2\). Option \(5n+2\) gives 22 when \(n=4\), so it is incorrect. Exam tip: when two terms are given, first find \(d\) by dividing the difference of the terms by the difference of their indices.

Direct answer: Option A, \\(a_n=6\cdot2^{n-1}\\). This is a geometric sequence because each term is obtained by multiplying the previous term by 2: 6, then 12, then 24, then 48. For a geometric sequence, the nth term is \\(a_n=a_1r^{n-1}\\), where \\(a_1\\) is the first term and \\(r\\) is the common ratio. Here \\(a_1=6\\) and \\(r=2\\), so \\(a_n=6\cdot2^{n-1}\\). Check: for n=1 it gives 6; for n=2 it gives 12; for n=3 it gives 24. Option A is therefore correct. Option B, \\(6n\\), describes linear growth and gives 12 at n=2 but 18 at n=3, not 24. Option C gives 2 at n=1, so it misses the first term. Option D gives 12 at n=1, so it also starts incorrectly. The exponent is n-1 because the first term has undergone zero multiplications. Memory cue: geometric nth term = first term times ratio to the power n minus 1.

Given
\(a_n=2n^2+3n-4\). Thus,
\(a_{10}=2(10)^2+3(10)-4=226\) and
\(a_7=2(7)^2+3(7)-4=115\). Therefore,
\(a_{10}-a_7=226-115=111\). A value such as 118 can result from an error while evaluating the squared term. In an exam, calculate the two terms separately before subtracting.

The terms are \(2^2,3^2,4^2,5^2,6^2,\ldots\). Since the first term corresponds to \(n=1\) and is \(2^2\), the \(n\)th term is \((n+1)^2\). The expression \(n^2+1\) gives \(5\) for the second term, whereas the actual second term is \(9\). Exam tip: test a proposed formula by substituting \(n=1\) and \(n=2\).

Given \(a_n=9-2n\), put \(a_n=-31\): \(9-2n=-31\). Thus, \(-2n=-40\), so \(n=20\). Therefore, the 20th term is \(-31\). The 19th term is \(9-2(19)=-29\), so it is not correct. Exam tip: To find a particular term number, substitute the given term value for \(a_n\) and solve for \(n\).

The first term is (10) and the difference is (-3), so (a_n=10+(n-1)(-3)=13-3n). A decreasing sequence has a negative difference.

Given \(a_n=4n+1\), substitute the complete index \(2m\) for \(n\): \(a_{2m}=4(2m)+1=8m+1\). Hence, option A is correct. \(4m+1\) would be obtained for the index \(m\), not \(2m\). Exam tip: Always substitute a compound index in brackets.

The sequence rule is linear: each term is obtained by multiplying its position by 7 and then subtracting 5. To find the change from one term to the next, first replace n by n+1. This gives \(a_{n+1}=7(n+1)-5=7n+2\). Now subtract the original term: \(a_{n+1}-a_n=(7n+2)-(7n-5)=7\). The variable terms cancel completely.

Therefore, option A, 7, is correct. The result can also be understood without expansion: increasing n by 1 increases \(7n\) by 7, while the constant \(-5\) does not change. Options 5 and 2 confuse the constant or the new expression with the difference. The supplied answer and explanation correctly identify the constant first difference of this linear sequence.

The consecutive differences are 5, 7, 9, and 11, increasing by 2 each time, so the rule is quadratic. Substituting \(n=1,2,3\) in \(n^2+2n\) gives 3, 8, and 15 respectively; hence the nth term is \(n^2+2n\). The closest distractor, \(n^2+n+1\), gives 7 when \(n=2\), not 8. Exam tip: substitute the first two or three values of \(n\) to verify an nth-term formula quickly.