Sequences

Since (\frac{10\times13}{2}=65), it is the (10)th term. Exam tip: substitute options directly into the formula.

The differences between consecutive terms are 3, 4, 7, and 12. Their second differences are 1, 3, and 5, so the next second difference is 7. Hence, the next first difference is 12+7=19, and the next term is 28+19=47. If 45 were chosen, the difference would be only 17, not the required 19. Exam tip: For such sequences, list first differences and then second differences to identify the pattern.

The governing concept is recognizing an alternating or interleaved sequence. Separate the odd and even positions. At odd positions we have 1, 2, 3, 4, ...; therefore the term at position 2k - 1 is k. At even positions we have 4, 8, 16, 32, ...; this is a geometric progression with first term 4 and ratio 2, so the term at position 2k is 4 multiplied by 2^(k - 1). Since 10 = 2(5), the 10th term is the fifth even-position term: 4, 8, 16, 32, 64. Thus option B is correct. Options A, C, and D correspond to taking the fourth or sixth even-position term or doubling one time too many.

For \(a_n=n^2+1\), the first difference is \(2n+1\), so it changes with \(n\), while the second difference is the constant \(2\). A linear sequence has constant first differences instead. Exam tip: check second differences when an \(n^2\) term appears.

Here the useful pattern is not in the terms themselves but in the differences between consecutive terms. From 2 to 7 the increase is 5, from 7 to 17 it is 10, from 17 to 32 it is 15, and from 32 to 52 it is 20. These differences increase by 5 each time. The next difference must therefore be 25. Adding that difference to the last known term gives 52 plus 25 equals 77, so option C is correct.

The calculation should be performed in two stages: first extend the difference pattern, then apply it to the last term. The difference list is 5, 10, 15, 20, 25, so the next term is 52 + 25. This equals 77. A value of 72 would add only 20 again, while 75 or 80 does not follow the next difference in the stated pattern. The supplied answer C is therefore consistent, and its explanation correctly emphasizes that recognizing the successive differences is the key step.

The governing concept is a recursive difference pattern. The listed successive differences are -3, -6, -9, and -12; each difference decreases by 3, so the next difference must be -15. The last displayed term is 90. Applying the next difference gives 90 + (-15) = 75, or equivalently 90 - 15 = 75. Thus option A is correct. The sequence is not an arithmetic progression because its term-to-term differences are not constant; instead, the differences themselves form an arithmetic progression. Options B, C, and D would correspond to subtracting 12, 9, or 6 again, rather than continuing the stated difference pattern by subtracting the next multiple of 3.

Odd positions are (4,8,16,\ldots), so the next odd-position term is (32). Exam tip: separate odd and even positions in alternating sequences.

Apply the recurrence step by step: \(a_2=3(2)-1=5\), \(a_3=3(5)-1=14\), and \(a_4=3(14)-1=41\). Therefore, \(a_5=3(41)-1=122\). The value 116 can result from an error in multiplication or subtraction in the final step. Exam tip: write each intermediate term before finding the required term.

The terms are (1,3,9,21,41), so (a_5=41). Exam tip: write positions along with values to avoid recursive mistakes.

The general term is (\frac{n+1}{2n+1}), so the (12)th term is (\frac{13}{25}). Exam tip: observe numerator and denominator patterns separately.

Given \(a_n=2^n-n\), substitute \(n=8\): \(a_8=2^8-8=256-8=248\). Therefore, 248 is the correct option. A value such as 240 may result from evaluating the power incorrectly or subtracting at the wrong step. Exam tip: calculate \(2^8=256\) first, then subtract 8.

(a_{11}=275) and (a_{12}=324), so (11) terms are less than (300). Exam tip: check the two terms near the boundary.

The difference between the second and third terms is 15, so x = 12 + 15 = 27. Checking it, 50 - 27 = 23, which matches the next given difference. Hence, 27 is correct; if x were 30, the next difference would be 20, not 23. Exam tip: after finding a missing term, verify it using the differences on both sides.

Use the stated successive differences in order. Starting from 171, the next term is obtained by adding -22, so x = 171 - 22 = 149. Checking the final step gives 149 + (-27) = 122, which agrees with the last given term. Therefore option A is correct; the other options fail this verification or mishandle the negative differences.

The rule is: next term = twice the previous term + 2. Hence, the term after 62 is \(2\times 62+2=126\). Although 124 is twice 62, it does not include the required addition of 2. Exam tip: in a recursive sequence, apply the stated rule directly to the last given term.

The successive terms are multiplied by 2, 3, 4, and 5: \(2\times2=4\), \(4\times3=12\), \(12\times4=48\), and \(48\times5=240\). Therefore, the next multiplier is 6, so the next term is \(240\times6=1440\). Option 1200 would repeat multiplication by 5, which does not follow the increasing multiplier pattern. Exam tip: divide consecutive terms to identify a changing-multiplier pattern.

The governing concept is obtaining and adding the required terms of a sequence. The pattern is n² + 1: for n = 1, 2, 3, 4, 5 it gives 2, 5, 10, 17, 26. Therefore the sixth term is 6² + 1 = 37. The first six terms are consequently 2, 5, 10, 17, 26, and 37. Adding them gives 2 + 5 + 10 + 17 + 26 + 37 = 97. Hence option C is correct. A quick check using the formula gives the same result: the sum of n² from 1 to 6 is 91, and adding 1 for each of the six terms gives 91 + 6 = 97. Options A, B, and D result from omitting or miscalculating a term.

In B, \(a_{n+1}-a_n=[3(n+1)-2]-(3n-2)=3\) for every \(n\), so it is an AP. For \(n^2+1\), the difference changes. Exam tip: compare consecutive differences.

For \(a_n=n^2+1\), the terms are 2, 5, 10, 17. Its first differences are 3, 5, 7, while the second differences are 2, 2. A linear sequence has constant first differences instead. Exam tip: constant second differences indicate a quadratic sequence.

The general term is (\frac{1}{3n}), and (\frac{1}{51}<\frac{1}{50}). So the first such term is the (17)th.