First, identify the pattern: 2, 6, 18, 54, ... This is a geometric sequence with first term 2 and common ratio 3 (since 6/2 = 3, 18/6 = 3, etc.). The general term is aₙ = 2 × 3^(n-1). Let's find which term equals 1458: 2 × 3^(n-1) = 1458, so 3^(n-1) = 729 = 3⁶, thus n-1 = 6, so n = 7. Therefore, 1458 is the 7th term. The original sequence starts: a₁ = 2, a₂ = 6, a₃ = 18, a₄ = 54, a₅ = 162, a₆ = 486, a₇ = 1458. If three terms are removed and we're left with 2, _, 1458, then the missing term must be one of the original terms that wasn't removed. Since we start with 2 (the 1st term) and end with 1458 (the 7th term), and three terms are removed, we have 7 - 3 = 4 remaining terms total. The pattern 2, _, 1458 shows 3 of the 4 remaining terms, so there's one missing term shown as _. Given that terms were removed, the missing term could be a₅ = 162. Let's verify: if we remove a₂ = 6, a₃ = 18, a₄ = 54, we get 2, 162, 486, 1458, and the pattern 2, _, 1458 with the blank being 162 makes sense if 486 was also removed. Actually, if three terms are removed to leave exactly 2, _, 1458, then we need exactly these three terms remaining. Working backwards: we need a term that, when multiplied by some power of 3, gives 1458, and when divided by some power of 3, gives 2. Since 1458/2 = 729 = 3⁶, the missing term should be 2 × 3ᵏ for some k such that this term × 3ʲ = 1458. If the missing term is 162 = 2 × 3⁴, then 162 × 3² = 162 × 9 = 1458 ✓. Also, 2 × 3² = 18, and 18 × 3² = 162 ✓. So the progression 2 → 162 → 1458 involves multiplying by 3⁴ then by 3², which suggests some intermediate terms were removed. Choice A (162) is correct.

Let's set up the sequence: a₁ = x, a₂ = x + 3, a₃ = a₁ + a₂ = x + (x + 3) = 2x + 3, a₄ = a₂ + a₃ = (x + 3) + (2x + 3) = 3x + 6, a₅ = a₃ + a₄ = (2x + 3) + (3x + 6) = 5x + 9. Given that a₅ = 34, we have: 5x + 9 = 34, so 5x = 25, thus x = 5. Therefore: a₁ = 5, a₂ = 8, a₃ = 13, a₄ = 21, a₅ = 34. The third term is 13. Choice B (8) is the second term. Choice C (10) might result from incorrectly calculating 2x + 3 with x = 5 as 2(5) = 10, forgetting the +3. Choice D (11) might result from an arithmetic error in solving for x or calculating 2x + 3.

Let's verify the pattern first: a₁ = 12, a₂ = 8, a₃ = (12+8)/2 = 10, a₄ = (8+10)/2 = 9, a₅ = (10+9)/2 = 9.5, a₆ = (9+9.5)/2 = 9.25, a₇ = (9.5+9.25)/2 = 9.375, a₈ = (9.25+9.375)/2 = 9.3125. The sequence appears to be converging. For a sequence where aₙ = (aₙ₋₁ + aₙ₋₂)/2, if it converges to a limit L, then eventually aₙ ≈ aₙ₋₁ ≈ aₙ₋₂ ≈ L. So L = (L + L)/2 = L, which is satisfied by any value. To find the actual limit, we can use the fact that for this type of averaging sequence, the limit is (2a₂ + a₁)/3. With a₁ = 12 and a₂ = 8: L = (2(8) + 12)/3 = (16 + 12)/3 = 28/3 ≈ 9.33. Let's verify this makes sense with our calculated terms: 9.25, 9.375, 9.3125, ... These are indeed approaching 9.33. Choice A (9.0) is too low. Choice C (9.5) is a₅ but not the limit. Choice D (10.0) is a₃ but not the limit. Choice B (9.33) is correct.

First, let's verify the pattern: 5 → 2(5)+1 = 11 → 2(11)+1 = 23 → 2(23)+1 = 47 → 2(47)+1 = 95 → 2(95)+1 = 191 → 2(191)+1 = 383. The complete sequence is: 5, 11, 23, 47, 95, 191, 383, ... The remaining sequence after removing two consecutive terms is 5, 23, _, 383. From our complete sequence, we can see that after 5 comes 11, then 23, then 47, then 95, then 191, then 383. Since we have 5, 23, _, 383 and two consecutive terms were removed, we need to determine which two. The gap between 23 and 383 in the original sequence includes 47, 95, 191. Since two consecutive terms were removed and we're left with a gap that shows _, 383, the two removed terms must be consecutive terms that came before the missing term. Looking at the pattern: if we have 5, then skip to 23 (so 11 was removed), then we have the missing term, then 383. Since 383 comes after 191 in the original sequence, and we need exactly one more removal, the two consecutive removed terms must be 11 and 47, leaving 5, 23, 95, 191, 383. But this doesn't match the given pattern. Let me reconsider: if the remaining sequence is 5, 23, _, 383, then working backwards from 383: the term before 383 is 191 (since 2(191)+1 = 383). So the missing term is 191. Choice A is correct. Choices B and C (95 and 47) are terms from the original sequence but don't fit the position. Choice D (159) doesn't appear in the original sequence at all.