nth term

This is an arithmetic progression because each term decreases by 4. Here, \(a=35\), \(d=-4\), and \(n=10\). Therefore, \(a_{10}=a+(n-1)d=35+9(-4)=35-36=-1\). Hence, -1 is correct. Getting 1 is a common error caused by applying the common difference incorrectly instead of using the 9 gaps before the 10th term. Exam tip: for the \(n\)th term, always use \(n-1\) common differences.

Given (a_n=5n^2-4n), substitute n=5: (a_5=5(5)^2-4(5)=5×25-20=125-20=105). Hence, 105 is the correct answer. An answer such as 100 may result from an error while squaring or multiplying. Exam tip: after substitution, evaluate powers first, then multiplication, and finally subtraction.

The successive differences are \(6,8,10\), whose differences are constant at \(2\); therefore, the term should contain an \(n^2\) part. For \(a_n=n^2+3n\), we get \(a_1=4\), \(a_2=10\), \(a_3=18\), and \(a_4=28\). The close distractor \(n^2+4\) gives 5 as its first term, so it is incorrect. Exam tip: test a proposed nth-term formula by substituting \(n=1\) and \(n=2\).

Given \(a_n=4n+18\) and \(a_n=70\), we get \(4n+18=70\). Thus, \(4n=52\), so \(n=13\). Therefore, 70 is the 13th term of the sequence. If \(n=12\), the term is \(4(12)+18=66\), not 70. Exam tip: To find a term’s position, substitute its value for \(a_n\) and solve the equation for \(n\).

The common ratio is \(\frac{1}{3}\), and \(a_5=120\left(\frac{1}{3}\right)^4=\frac{40}{27}\). Use powers carefully with a fractional ratio.

Given \(a_n=18-6n\), substitute \(n=5\) to find the fifth term: \(a_5=18-6(5)=18-30=-12\). Therefore, option C is correct. A nearby distractor such as \(-9\) can result from multiplying \(6\times5\) incorrectly. Exam tip: after substituting the value of \(n\), perform multiplication before subtraction.

The first differences are \(9,11,13\), whose differences are constant at \(2\); hence the pattern is quadratic. For \(a_n=n^2+6n\), substituting \(n=1,2,3,4\) gives \(7,16,27,40\), respectively. \(9n-2\) fits only the first two terms and gives \(25\) for the third term. Exam tip: when second differences are constant, test a rule containing \(n^2\).

Putting \(n=4\), \(a_4=4\cdot5^{4-1}=4\cdot5^3=4\cdot125=500\). Hence, the correct answer is 500. The value 400 would result from incorrectly using an exponent of 2. Exam tip: when substituting into \(a_n\), simplify the exponent \(n-1\) carefully first.

This is an arithmetic progression with first term 17 and common difference 5. Therefore, its nth term is \(a_n=17+(n-1)\times5=5n+12\). Putting \(a_n=87\) gives \(5n+12=87\), so \(5n=75\) and \(n=15\). At position 14, the term would be 82, not 87. Exam tip: first identify the first term and common difference, then use \(a_n=a+(n-1)d\).

Given \(a_n=n^2+4n-2\), substitute \(n=6\): \(a_6=6^2+4\times6-2=36+24-2=58\). Therefore, 58 is correct. The value 60 would result if the final \(-2\) were omitted. Exam tip: after substituting a value in an nth-term rule, follow the order of operations carefully.

The first term of this GP is 3 and the common ratio is 5. Its terms are 3, 15, 75, 375, 1875. Therefore, 1875 is the fifth term. The sixth term would be 1875 × 5 = 9375, so the sixth-term option is not correct. Exam tip: To find a term’s position, list the terms in order or use \(a_n=ar^{n-1}\).

The governing concept is the nth term of an arithmetic progression, aₙ = a₁ + (n − 1)d. The first term is a₁ = 9, and the common difference is d = 19 − 9 = 10; the later differences also equal 10. Therefore, aₙ = 9 + (n − 1)10 = 9 + 10n − 10 = 10n − 1. Option A is correct. Verification gives a₁ = 10(1) − 1 = 9, a₂ = 19, a₃ = 29, and a₄ = 39. Option B gives 11 as the first term, option C has coefficient 9 and therefore difference 9 rather than 10, and option D has difference 1. Thus only option A preserves both defining features of the sequence.

To find the fifth term, substitute \(n=5\): \(a_5=\frac{7(5)-2}{3}=\frac{35-2}{3}=\frac{33}{3}=11\). Therefore, 11 is the correct option. Option 10 may result from an error in subtraction or division. Exam tip: In an \(a_n\) formula, substitute the required value of \(n\) and simplify step by step.

The sequence is 4, 16, 36, 64, and each term can be viewed as four times a square. The first term is \\(4\\times1^2\\), the second is \\(4\\times2^2\\), the third is \\(4\\times3^2\\), and the fourth is \\(4\\times4^2\\). Thus the number inside the square is exactly the position number. Recognizing this structure is more reliable than guessing from differences alone.

For the nth position, replace the position number by \\(n\\). The general term is therefore \\(a_n=4n^2\\). Checking it confirms the pattern: \\(4(1)^2=4\\), \\(4(2)^2=16\\), \\(4(3)^2=36\\), and \\(4(4)^2=64\\). Hence option A is correct. Option B gives half the required values, while the linear expression in option C cannot produce this square-based growth.

Given \(a_n=12n+5\), \(a_7=12\times7+5=89\) and \(a_2=12\times2+5=29\). Therefore, \(a_7+a_2=89+29=118\), so option A is correct. A value such as 116 can result from a small arithmetic error while adding or multiplying. Exam tip: evaluate each required term separately before adding them.

The terms are (9,27,81,243,729,2187), so (2187) is the sixth term. In a geometric progression, multiply by the same ratio each time.

To find the sixth term, substitute n=6 in the rule: \(a_6=3(6)^2+5(6)=3\times36+30=108+30=138\). Therefore, 138 is correct. A value such as 128 may result from an error in squaring or multiplication. Exam tip: evaluate the exponent first, then multiply, and finally add.

The first term is (50) and the difference is (-7), so (a_n=50+(n-1)(-7)=57-7n). In a decreasing sequence, take the difference as negative.

The governing concept is the general term of an arithmetic sequence. The consecutive differences are 11−7 = 4, 15−11 = 4, and 19−15 = 4, so the common difference is d = 4 and the first term is a₁ = 7. Applying aₙ = a₁ + (n−1)d gives aₙ = 7 + (n−1)4 = 7 + 4n − 4 = 4n + 3. Therefore option A is correct. Substitution verifies the rule: n = 1 gives 7, n = 2 gives 11, n = 3 gives 15, and n = 4 gives 19. Option B has an incorrect coefficient and constant; options C and D also fail to reproduce the stated first term and the constant difference.

The governing concept is identifying a pattern and expressing it as a general term. The given numbers are consecutive cubes: 8 = 2³, 27 = 3³, 64 = 4³, and 125 = 5³. Since the first term corresponds to base 2 rather than base 1, the base for the nth term is n + 1. Thus aₙ = (n + 1)³, so option A is correct. The rule can be verified at more than one position: for n = 1, it gives (1 + 1)³ = 8; for n = 2, it gives 3³ = 27; and for n = 4, it gives 5³ = 125. Option B is not equivalent to the correct expression because expanding the latter gives n³ + 3n² + 3n + 1. Options C and D also fail these checks.

To find when a term is zero, set the explicit rule equal to zero: aₙ = 15 − 2n = 0. Solving gives 2n = 15 and n = 15/2 = 7.5. Sequence indices are positive integers, so there is no integral term number equal to 7.5. Consequently, no term of this sequence is exactly zero, and option A is correct. Checking nearby valid indices confirms this: a₇ = 15 − 14 = 1 and a₈ = 15 − 16 = −1, so the sequence passes from positive to negative without taking zero at an integer index. Options B and C identify neighbouring terms but neither is zero, while D is unrelated.

Direct answer: option A, 6n − 2. In an arithmetic sequence, aₙ = a₁ + (n − 1)d, where d is the constant difference. From the fifth term to the eleventh term there are 11 − 5 = 6 equal steps. The total increase is 64 − 28 = 36, so 6d = 36 and d = 6. Now use the fifth term: a₅ = a₁ + 4d, so 28 = a₁ + 24 and a₁ = 4. Therefore aₙ = 4 + (n − 1)6 = 4 + 6n − 6 = 6n − 2. Checking gives a₅ = 30 − 2 = 28 and a₁₁ = 66 − 2 = 64. Option B has the wrong constant term and gives 34 at n = 5. Option C has common difference 5, not 6. Option D has common difference 7 and fails the data. Common confusion: between a₅ and a₁₁ there are six gaps, not five.

Direct answer: option B, 5n + 1. For an arithmetic sequence, aₙ = a₁ + (n − 1)d. The eighth term is seven common differences after the first term, so a₈ = a₁ + 7d. Substitute the known values: 41 = a₁ + 7(5) = a₁ + 35. Therefore a₁ = 6. Now substitute into the general formula: aₙ = 6 + (n − 1)5 = 6 + 5n − 5 = 5n + 1. Check at n = 8: 5(8) + 1 = 41, exactly as given. Option A gives 46 at n = 8, so its constant term is too large. Option C gives 36, so its constant term is too small. Option D gives 75 and is clearly inconsistent. The coefficient of n in an arithmetic sequence is the common difference, so the coefficient 5 in the correct answer is also a useful check. Do not use 8d; the first term is seven steps before the eighth term.

Direct answer: option C, the 8th term. We need the index n for which aₙ equals 48, so set n(n − 2) = 48. Expand: n² − 2n − 48 = 0. Factor: (n − 8)(n + 6) = 0. Thus n = 8 or n = −6. A sequence position is a positive integer, so n = −6 is not an acceptable term number. Therefore 48 is the eighth term. Direct checking gives a₈ = 8(8 − 2) = 8 × 6 = 48. The distractors can be checked directly: a₆ = 6 × 4 = 24, a₇ = 7 × 5 = 35, and a₉ = 9 × 7 = 63. None equals 48. The important idea is that a formula gives the value from an index, while this question asks us to reverse the process and find the index. Memory cue: equate the formula to the target, solve, then reject any non-positive index.

The governing concept is evaluating an explicit formula for a sequence at two specified indices and then subtracting the resulting terms. Substitute n = 8 throughout the formula: a₈ = 3(8²) − 2(8) + 1 = 3 × 64 − 16 + 1 = 177. Next substitute n = 5: a₅ = 3(5²) − 2(5) + 1 = 3 × 25 − 10 + 1 = 66. Therefore, a₈ − a₅ = 177 − 66 = 111, so option D is correct. It is important to calculate both terms separately because the expression asks for the difference of two sequence values, not the value of the formula at 8 − 5. Options A, B, and C result from incorrect substitution or arithmetic.