Find the missing term of the geometric sequence.4, ____ , 5.76...

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txmedteach | High School Teacher | (Level 3) Associate Educator

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A geometric sequence is a sequence that takes the following form:

`a_n = a*r^(n-1)`

Here, `a` is the initial term, `r` is a ratio term that relates each term to the next, and n is the number term. Notice that `n >=1`, though some may define the same series with `n>=0`.

Therefore, each of the terms above can be related to the above sequence:

`4 = ar^0` 

`? = ar^1`

`5.76 = ar^2`

Using the first term, we can clearly see that `a = 4`. We can then use the third term to find the ratio:

`5.76 = 4r^2`

Divide both sides by 4:

`1.44 = r^2`

Now, subtract 1.44 from both sides. We could take the square root, but that won't give us the full picture!

`0 = r^2-1.44`

Finally, factor:` `

`0 = (r-1.2)(r+1.2)`

Solving this, we see two possible values for `r`:

`r = +-1.2`

Therefore, we have two possible values for the second term:

`a_2 = ar = +-4*1.2 = +-4.8`

There is your final answer! Again, we cannot say based on the given information whether the term is positive or negative, so we are left with two answers.