Question 1206538: Geometric Sequence in which T5 = 8T2
T4 + T6 = 240
Find the Sequence.
Found 2 solutions by greenestamps, math_tutor2020:
Answer by greenestamps(13200)   (Show Source):
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
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:
6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, ...
Start at 6. Double each term to get the next term.
--------------------------------------------------------------------------
Verification:
Focus on the terms T2 and T5 highlighted in blue
6, 12, 24, 48, 96, 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, 48, 96, 192, 384, 768, 1536, 3072, ...
So,
T4 + T6 = 240
48 + 192 = 240
240 = 240
The second requirement is verified.
The answer is fully confirmed.
|
|
|
