Question 1206538
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T1 = first term
T2 = second term
T3 = third term
etc


To go from term to term in a geometric sequence, we multiply by a common ratio r.
T2 = r*T1
T3 = r*T2 = r*(r*T1) = r^2*T1
T4 = r*T3 = r*(r^2*T1) = r^3*T1
T5 = r*T4 = r*(r^3*T1) = r^4*T1
etc


To avoid a sequence of nothing but 0s, we require that r and T1 are both nonzero. 
By extension it will mean r*T1 is also nonzero.
This will allow us to divide both sides by r*T1 as shown in the next section.


T5 = 8*T2
r^4*T1 = 8*r*T1
r^3 = 8 .................... divide both sides by r*T1
r = cubeRoot(8)
r = 2


Now use the second given equation.
T4 + T6 = 240
r^3*T1 + r^5*T1 = 240
2^3*T1 + 2^5*T1 = 240
8*T1 + 32*T1 = 240
40*T1 = 240
T1 = 240/40
T1 = 6 is the first term of the geometric sequence.


The geometric sequence is therefore:
<font color=red>6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, ...</font>
Start at 6. Double each term to get the next term.


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Verification:


Focus on the terms T2 and T5 highlighted in <font color=blue>blue</font>
6, <font color=blue>12</font>, 24, 48, <font color=blue>96</font>, 192, 384, 768, 1536, 3072, ...


Then notice:
T5 = 8*T2
96 = 8*12
96 = 96
The first requirement is verified.


Now let's highlight the 4th and 6th terms.
6, 12, 24, <font color=blue>48</font>, 96, <font color=blue>192</font>, 384, 768, 1536, 3072, ...
So,
T4 + T6 = 240
48 + 192 = 240
240 = 240
The second requirement is verified.
The answer is fully confirmed.
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