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