nth term

The governing pattern is the sequence of perfect squares: the nth term is \(a_n=n^2\). Thus the required difference is \(a_{15}-a_{14}=15^2-14^2\). Using the difference-of-squares identity \(p^2-q^2=(p-q)(p+q)\), we get \((15-14)(15+14)=1\times29=29\). Therefore option C is correct. Direct calculation also confirms it: \(15^2=225\) and \(14^2=196\), so \(225-196=29\). Options A, B, and D arise from an incorrect square or subtraction and do not equal the difference of the two specified terms.

The sequence is an arithmetic progression: the first term is \(a=-5\), and the common difference is \(d=4\), since each term increases by 4. Use the general term formula \(a_n=a+(n-1)d\). For \(n=25\), \(a_{25}=-5+(25-1)4=-5+24\times4=-5+96=91\). Therefore option B is correct. The negative first term must be included in the calculation. Option A effectively misses one increment of 4, while options C and D add too many increments or mishandle the starting value.

The governing concept is evaluation of an explicit, or general, rule for a sequence. Substitute n = 7 and n = 5 separately into aₙ = n³ − n. We get a₇ = 7³ − 7 = 343 − 7 = 336, and a₅ = 5³ − 5 = 125 − 5 = 120. Therefore, a₇ − a₅ = 336 − 120 = 216, so option B is correct. A useful algebraic check is 7³ − 7 − (5³ − 5) = 343 − 7 − 125 + 5 = 216. The other options arise from an incorrect cube, forgetting the subtraction of n, or subtracting the terms incorrectly. Evaluating each term carefully avoids those errors.

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 finding terms from an explicit nth-term formula. The expression must be evaluated separately at the two requested indices, while preserving the parentheses. For n = 11, a_11 = 11(11 + 2) = 11 × 13 = 143. For n = 8, a_8 = 8(8 + 2) = 8 × 10 = 80. Subtracting the actual term values gives a_11 − a_8 = 143 − 80 = 63. Therefore option A is correct. A frequent mistake is to subtract the indices, or to interpret n(n + 2) as n² + 2; neither follows the stated formula. Options B, C and D are therefore not supported by the required substitution and subtraction.

The governing concept is using an explicit nth-term rule to identify the position of a given value. We set the formula equal to 148: n^2 + 5n - 2 = 148, so n^2 + 5n - 150 = 0. Factoring gives (n + 15)(n - 10) = 0, so n = 10 or n = -15. Since a term number must be a positive integer, n = 10 is the valid solution. Direct substitution confirms it: a_10 = 10^2 + 5(10) - 2 = 100 + 50 - 2 = 148. Therefore option D is correct. The negative root is rejected because sequence positions are counted as 1, 2, 3, and so on; options A, B, and C do not produce 148.

The governing concept is identifying a term number from an explicit, or general, rule. We need an integer n for which a_n = n² + 2n equals 168. Substitute the possible position n = 12: a_12 = 12² + 2(12) = 144 + 24 = 168. Therefore, 168 is the 12th term, so option C is correct. Checking nearby positions confirms the choice: a_11 = 121 + 22 = 143 and a_13 = 169 + 26 = 195, so neither adjacent position gives 168. The other options result from using an incorrect position or arithmetic.

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 a term position from an nth-term formula. We seek n such that 4n² − 1 = 575. Adding 1 to both sides gives 4n² = 576. Dividing by 4 gives n² = 144, so the positive term position is n = 12; a sequence position cannot be negative. Direct substitution confirms this: a_12 = 4(12²) − 1 = 4 × 144 − 1 = 576 − 1 = 575. Therefore option C is correct. The nearby choices give a_11 = 483 and a_13 = 675, so they cannot produce 575. The negative square root is rejected because term numbers are positive integers.

The governing concept is locating a term using an explicit product-form rule. We need n(n + 3) = 130. Testing the listed positions is efficient: for n = 10, a_10 = 10(10 + 3) = 10 × 13 = 130. Thus 130 is the 10th term, making option C correct. Algebraically, n² + 3n − 130 = 0, which factors as (n + 13)(n − 10) = 0. The possible roots are n = 10 and n = −13, but a sequence position must be positive, so only n = 10 is valid. The other options yield 88, 108, and 154 respectively, so they are not equal to 130.

The governing concept is recognizing the square-number sequence. The terms are 1², 2², 3², 4², and so on, so its general rule is a_n = n². Therefore a_11 = 11² = 121 and a_12 = 12² = 144. Adding these required terms gives a_11 + a_12 = 121 + 144 = 265. Hence option C is correct. The nearby choices can arise from squaring only one position, using the wrong position, or making an addition error. It is important not to interpret a_11 + a_12 as the term with index 23; it means the values of the 11th and 12th terms must be calculated separately and then added.

The governing concept is finding the first sequence term that crosses a specified boundary. Because the rule a_n = n² + 2n increases for positive n, it is sufficient to check consecutive positions near 100. For n = 9, a_9 = 9² + 2(9) = 81 + 18 = 99, which is not greater than 100. For n = 10, a_10 = 10² + 2(10) = 100 + 20 = 120, which is greater than 100. Therefore the first qualifying term is the 10th term, so option B is correct. The 9th term fails the strict inequality, while later options are not the first because n = 10 already satisfies it.

The governing pattern is the sequence of perfect cubes: 1 = 1^3, 8 = 2^3, 27 = 3^3, and 64 = 4^3. Thus the general term is a_n = n^3. The ninth term is a_9 = 9^3 = 729, and the tenth term is a_10 = 10^3 = 1000. Therefore a_10 - a_9 = 1000 - 729 = 271. Equivalently, 10^3 - 9^3 = (10-9)(10^2+10×9+9^2) = 271. Hence option A is correct; nearby options reflect arithmetic errors in cubing or subtraction.

The governing concept is an explicit, or general, rule: it must produce the term directly from its position n. Test the proposed rule at n=1, 2, 3 and 4. For option B, (2(1)+3)/5=1, (2(2)+3)/5=7/5, (2(3)+3)/5=9/5, and (2(4)+3)/5=11/5. Thus every displayed term agrees, so option B is correct. Option A gives 1 at n=1 but 6/5 at n=2; option C gives 7/3 at n=1; and option D gives 1, 3, 5, 7, not the given fractional pattern. Therefore B is the only unambiguous rule.

An explicit rule must reproduce each term from its position. The first differences are 12, 20 and 28, so the second differences are 8 and 8. Constant second differences suggest a quadratic expression. Substitute n=1, 2, 3 and 4 into option B: 4(1)^2+1=5, 4(2)^2+1=17, 4(3)^2+1=37, and 4(4)^2+1=65. Hence option B exactly generates the sequence. Option A is linear and gives 9 at n=2; option C gives 20 at n=1; and option D gives 5, 17, 29, 41, so it fails from the third term. Therefore B is correct.

The relevant concept is recognizing the pattern of perfect powers and connecting each term with its position. The displayed values can be written as 1=1^3, 8=2^3, 27=3^3 and 64=4^3. Since the nth term is the cube of n, the explicit rule is a_n=n^3, so option C is correct. Option A gives squares, producing 1, 4, 9, 16. Option B gives powers of 2, producing 2, 4, 8, 16, and option D starts with 8 because it uses (n+1)^3. Checking the first position is especially useful for distinguishing n^3 from (n+1)^3.

This is an arithmetic sequence because consecutive terms differ by the constant amount 6. Its nth-term formula is a_n=a_1+(n-1)d=7+(n-1)6, which simplifies to a_n=6n+1. For the 25th term, substitute n=25: a_25=6(25)+1=150+1=151. Therefore option C is correct. Option A corresponds to using an incorrect difference or index adjustment, option B is one less than the correct value, and option D is one term-step too high. The important principle is to use the same common difference for every step and to count 24 differences from the first term to the 25th term, not 25 differences.

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 positions. The terms are 27 = 3³, 64 = 4³, 125 = 5³, and 216 = 6³. The cube bases therefore begin at 3 when the position is n = 1, and increase by one as n increases. The base can be written as n + 2, so the general term is aₙ = (n + 2)³. Checking confirms that n = 1 gives 3³ = 27 and n = 4 gives 6³ = 216. Option A would start with 1, option B with 2, and option D does not preserve the cube pattern. Thus C is correct.

The governing concept is deriving a general term from a pattern of perfect squares. Rewrite the sequence as 2², 6², 10², and 14². The bases form an arithmetic sequence beginning at 2 and increasing by 4, so the nth base is 2 + (n − 1)4 = 4n − 2. Squaring this base gives aₙ = (4n − 2)². Substitution confirms the result: 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; C starts with 16; and D is not consistent. Hence B is correct.