Sequences

The governing concept is a sequence controlled by a second pattern: the successive differences are \(-5,-8,-11,-14\). These differences form an arithmetic progression with common difference \(-3\), so the next difference must be \(-14-3=-17\). Apply that difference to the last known term, 12: \(12+(-17)=12-17=-5\). Thus option C is correct. Option A would use an unjustified difference of \(-12\), while options B and D do not follow the continuing decrease by 3. Sign discipline is essential here: the next change is negative, so the term must decrease from 12. Treating 17 as positive would incorrectly give 29 rather than the required next term.

The governing concept is recognizing an alternating, or interleaved, sequence. Separate the odd-position and even-position terms. The odd positions are 1, 2, 3, 4, so the term at position 2k − 1 is k. The even positions are 4, 8, 16, 32, and so on; this is a geometric pattern with first term 4 and common ratio 2. Since position 10 is the fifth even position, its value is 4 × 2^(5 − 1) = 4 × 16 = 64. Equivalently, listing the terms gives the 8th term as 32 and the 10th term as 64. Therefore option B is correct. Option A is the 8th term, while C and D result from continuing the doubling too many times.

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.

The governing concept is a geometric sequence, where every term is obtained by multiplying the previous term by a constant common ratio. Here the ratio is −6/3 = −2; the next checks, 12/(−6) = −2 and −24/12 = −2, confirm the pattern. Therefore the nth term is a_n = 3(−2)^(n−1). For n = 9, a_9 = 3(−2)^8. Since (−2)^8 = 256 because the exponent is even, a_9 = 3 × 256 = 768. Thus option C is correct. The negative answers overlook the fact that an even power of a negative number is positive. The value 384 results from an arithmetic or exponent error and is only half of the required product.

The governing concept is identifying the pattern and then adding the required number of terms. The displayed sequence follows a_n = n² + 1: for n = 1, 2, 3, 4, and 5, this gives 2, 5, 10, 17, and 26. The sixth term is therefore a_6 = 6² + 1 = 37. The first six terms are 2, 5, 10, 17, 26, and 37, whose sum is 2 + 5 + 10 + 17 + 26 + 37 = 97. Thus option C is correct. A second check uses the square-sum formula: 1² + 2² + ... + 6² = 91, and adding 1 once for each of six terms gives 91 + 6 = 97. Other options arise from omitting a term or making an addition error.