Sequences

Correct answer: C. -5

Explanation: 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.

Correct answer: B. 64

Explanation: 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.

Correct answer: A. 75

Explanation: 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.

Correct answer: C. 768

Explanation: 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.

Correct answer: C. 97

Explanation: 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.