Question

Prove that the sequence tn = 6(1/2)ⁿ⁻¹ is geometric.

Answer 1

To prove the sequence is geometric, we showed that the ratio between consecutive terms is consistent. By comparing the first few terms, we found that each term is obtained by multiplying the previous term by the common ratio 1/2, confirming the sequence's geometric nature.

Step-by-step explanation:

To prove that the sequence tn = 6(1/2)n-1 is geometric, we must show that each term can be obtained by multiplying the previous term by a common ratio. In a geometric sequence, the ratio between consecutive terms is constant. The common ratio r can be found by dividing any term (after the first) by its preceding term.

Step-by-step verification:

1. Let's consider the first two terms of the sequence, t1 and t2.
2. The first term is t1 = 6.
3. The second term is t2 = 6(1/2)1 = 3.
4. Now, divide t2 by t1 to find the common ratio: r = t2/t1 = 3/6 = 1/2.
5. To ensure this ratio is consistent, look at t3 = 6(1/2)2 = 1.5 and divide by t2: r = t3/t2 = 1.5/3 = 1/2.
6. Since the ratio is consistent, we can conclude that the sequence tn = 6(1/2)n-1 is indeed geometric with a common ratio of 1/2.