Question

The first terms of an infinite geometric sequence, , are 2, 6, 18, 54, ... the first terms of a second infinite geometric sequence, , are 2, -6, 18, -54, ... the terms of a third sequence, , are defined as . (a) write down the first three non-zero terms of .

Answer 1

The third sequence is the sum of the first n terms of one of the geometric sequences defined in the question, and by applying a specific pattern, we can calculate its terms using the formula 2n². The first three non-zero terms of this sequence are 2, 8, and 18.

Step-by-step explanation:

The question involves geometric sequences and the sum of their terms. The first sequence is a standard geometric sequence with a common ratio of 3. The second sequence is similar but alternates in sign, giving it a common ratio of -3. The third sequence is defined as the sum of the first n terms of one of the sequences and using a pattern to generalize the sum.

To find the first non-zero terms of the third sequence, sequence C, we need to consider the sum of terms of the preceding sequences with alterations described (adding and subtracting certain terms) to derive a general formula, which in this case is 2n². Applying this formula, we can calculate the first three non-zero terms of sequence C by plugging in n = 1, 2, and 3, resulting in 2, 8, and 18, respectively.