100 Points!! Please help!

The first term of a geometric sequence is 4, and the sum of the first three terms is 28. Draw conclusions about the common ratio of this sequence, and use this information to find the explicit formula for the sequence. Then, use the explicit formula to find the seventh term of the sequence.

$\boxed{\begin{array}{l}\underline{\textsf{General form of the $n$th term of an geometric sequence}}\\\\a_n=ar^{n-1}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a_n$ is the $n$th term.}\\\phantom{ww}\bullet \;\textsf{$a$ is the first term.}\\\phantom{ww}\bullet\;\textsf{$r$ is the common ratio.}\\\phantom{ww}\bullet\;\textsf{$n$ is the position of the term.}\end{array}}$

Given that the first term in the sequence is 4, then:

$a_n=4r^{n-1}$

Given that the sum of the first three terms is 28, then:

$a_1+a_2+a_3=28\\\\\\4+4r^{2-1}+4r^{3-1}=28\\\\\\4+4r^{1}+4r^{2}=28\\\\\\4+4r+4r^{2}=28\\\\\\4(1+r+r^2)=4(7)\\\\\\1+r+r^2=7\\\\\\r^2+r-6=0$

Solve for r:

$r^2+3r-2r-6=0\\\\\\r(r+3)-2(r+3)=0\\\\\\(r+3)(r-2)=0\\\\\\r=-3,\;\;r=2$

So, we have two possible common ratios, r = -3 and r = 2.

If r = -3, then the sequence alternates signs:

$a_1=4\\\\a_2=4(-3)^1=-12\\\\a_3=4(-3)^2=36\\\\a_4=4(-3)^3=-108$

If r = 2, then:

$a_1=4\\\\a_2=4(2)^1=8\\\\a_3=4(2)^2=16\\\\a_4=4(2)^3=32$

The two possible explicit formulas for the sequence given the two found common ratios are:

$a_n=4(-3)^{n-1}\\\\a_n=4(2)^{n-1}$

To find the seventh term, we can substitute n = 7 into the formulas:

$a_7=4(-3)^{7-1}\\\\a_7=4(-3)^6\\\\a_7=4(729)\\\\a_7=2916\\\\\\\\a_7=4(2)^{7-1}\\\\a_7=4(2)^6\\\\a_7=4(64)\\\\a_7=256$

Therefore, the seventh terms are 2,916 and 256.