The second term of a geometric Progression is 12 more than the first term. If the common ratio is half of first term, find the third term.
a ar ar^2 ar^3 ar^4
ar - a = 12
r = a/2
so
a (a/2) - a = 12
a^2/2 - a - 12 = 0
a^2 - 2a - 24 = 0
(a-6)(a+4) = 0
a = 6 (or a =-4)
try a = 6, then r = 3
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6 18 54 .... yep, that works :)
try a = -4 then r = -2
-4 8 -16 well that seems to work too
To find the third term of a geometric progression, we need to determine the first term and the common ratio.
Let's assume that the first term of the geometric progression is "a", and the common ratio is "r".
From the given information, we know that the second term of the progression is 12 more than the first term. So, we can write:
Second term = First term + 12
ar = a + 12
We are also told that the common ratio is half of the first term. Therefore, we can write:
r = (1/2) * a
Now, we can substitute this value of r in the equation ar = a + 12:
(a/2) * a = a + 12
a^2/2 = a + 12
a^2 = 2(a + 12)
a^2 = 2a + 24
a^2 - 2a - 24 = 0
Now, we can solve this quadratic equation to find the value of "a". Once we obtain the value of "a", we can substitute it back into the equation r = (1/2) * a to obtain the value of the common ratio "r".
After determining the values of "a" and "r", we can calculate the third term as follows:
Third term = (First term) * (Common ratio)^2
By plugging in the values of the first term and the common ratio, we can find the third term of the geometric progression.