Question

Find the next four terms of the geometric sequence given the a1=2x r=x³

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Answer 1

Answer:
`2\text{x}^4, \ 2\text{x}^7, \ 2\text{x}^(10), \ 2\text{x}^(13)`

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Step-by-step explanation:

The first term is
`a_1 = 2\text{x}` and the common ratio is
`r = \text{x}^3`

To get the second term, we multiply the first term by that common ratio

`a_2 = a_1*r = 2\text{x}*\text{x}^3 = 2\text{x}^4`

Repeat a similar idea for the third term

`a_3 = a_2*r = 2\text{x}^4*\text{x}^3 = 2\text{x}^(4+3) = 2\text{x}^7`

and the fourth term is

`a_4 = a_3*r = 2\text{x}^7*\text{x}^3 = 2\text{x}^(7+3) = 2\text{x}^(10)`

The fifth term is

`a_5 = a_4*r = 2\text{x}^(10)*\text{x}^3 = 2\text{x}^(10+3) = 2\text{x}^(13)`

Something to notice: The exponents of the first five terms are: 1, 4, 7, 10, 13. The sequence of exponents is arithmetic even though the original underlying sequence is geometric.

The reason why the exponent sequence is arithmetic is because we keep multiplying by
`\text{x}^3`, and hence we keep adding 3 to each exponent to get the next exponent.

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To summarize, the first five terms are:

`2\text{x}, \ 2\text{x}^4, \ 2\text{x}^7, \ 2\text{x}^(10), \ 2\text{x}^(13)`

We will ignore the first term 2x since your teacher wanted you to find the next four terms after that first term (i.e. term2 through term5).