. The given sequence is not a geometric sequence. Explanation: A geometric sequence is defined by a constant ratio between consecutive terms. In this case, the sequence is defined by the formula a_n = 6n^2 + 5. To check if it’s geometric, we need to see if the ratio of consecutive terms is constant. Let’s take two consecutive terms, for example, a_1 and a_2: a_1 = 6(1)^2 + 5 = 6 + 5 = 11 a_2 = 6(2)^2 + 5 = 6(4) + 5 = 24 + 5 = 29 The ratio of a_2 to a_1 is: r = a_2 / a_1 = 29 / 11 Now let’s take a_2 and a_3: a_3 = 6(3)^2 + 5 = 6(9) + 5 = 54 + 5 = 59 The ratio of a_3 to a_2 is: r = a_3 / a_2 = 59 / 29 Since the ratio between consecutive terms is not constant (29/11 ≠ 59/29), the sequence is not geometric. Therefore, the correct answer is C.
The Correct Answer and Explanation is:
The given sequence is defined by the formula an=6n2+5a_n = 6n^2 + 5an=6n2+5, where nnn is the term number. To determine whether this sequence is a geometric sequence, we need to check if the ratio between consecutive terms is constant.
Step-by-Step Process:
- Find the first few terms of the sequence:
- For n=1n = 1n=1: a1=6(1)2+5=6+5=11a_1 = 6(1)^2 + 5 = 6 + 5 = 11a1=6(1)2+5=6+5=11
- For n=2n = 2n=2: a2=6(2)2+5=6(4)+5=24+5=29a_2 = 6(2)^2 + 5 = 6(4) + 5 = 24 + 5 = 29a2=6(2)2+5=6(4)+5=24+5=29
- For n=3n = 3n=3: a3=6(3)2+5=6(9)+5=54+5=59a_3 = 6(3)^2 + 5 = 6(9) + 5 = 54 + 5 = 59a3=6(3)2+5=6(9)+5=54+5=59
- Calculate the ratio between consecutive terms:
- The ratio between a2a_2a2 and a1a_1a1 is: r1=a2a1=2911≈2.636r_1 = \frac{a_2}{a_1} = \frac{29}{11} \approx 2.636r1=a1a2=1129≈2.636
- The ratio between a3a_3a3 and a2a_2a2 is: r2=a3a2=5929≈2.034r_2 = \frac{a_3}{a_2} = \frac{59}{29} \approx 2.034r2=a2a3=2959≈2.034
- Comparison of ratios:
- As shown above, the ratio between a2a_2a2 and a1a_1a1 is approximately 2.636, while the ratio between a3a_3a3 and a2a_2a2 is approximately 2.034. These ratios are not equal, which indicates that the sequence does not have a constant ratio between consecutive terms.
Conclusion:
Since the ratio between consecutive terms is not constant, this sequence is not a geometric sequence. A geometric sequence requires a constant ratio between consecutive terms, and since this condition is not met, the sequence does not qualify as geometric.
Therefore, the correct conclusion is that the sequence is not geometric.
