- The sequence 2,6,18,54,… follows a geometric pattern. If four terms are removed from the first seven terms of this sequence so that the remaining terms 2,,1458 still form a geometric sequence, what is the missing term?
Answer: 54
The sequence has first term 2 and common ratio 3, so its first seven terms are 2, 6, 18, 54, 162, 486, 1458, and 1458 is the 7th term. Removing four of these seven terms leaves three. For 2, __, 1458 to be geometric, the middle term m needs the same ratio on both sides: m ÷ 2 = 1458 ÷ m, so m² = 2 × 1458 = 2916 and m = 54. Check: 54 ÷ 2 = 27 and 1458 ÷ 54 = 27, so the common ratio is 27. The value 162 is a term of the original sequence, but 162 ÷ 2 = 81 while 1458 ÷ 162 = 9, so the ratios differ. Likewise 486 gives ratios of 243 and 3, and 18 gives ratios of 9 and 81.
- A sequence follows the pattern where each term is the sum of the two preceding terms, similar to the Fibonacci sequence. If the first term is x, the second term is x+3, and the fifth term is 34, what is the value of the third term?
Answer: 13
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.
- Consider the sequence where each term is the average of the two preceding terms: 12,8,10,9,9.5,9.25,… If this pattern continues indefinitely, what value does the sequence approach?
Answer: 9.33
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.
