The second term of a G.P is 12 more than the first term,given that the common ratio is half of the first term.find the fourt term of the G.P.
ar = a+12
r = a/2
a^2/2 = a+12
a^2 - 2a - 24 = 0
(a-6)(a+4)
so, a=6, r=3
or, a=-4, r=-2
6, 18, 54, 162, ...
or
-4, 8, -16, 32, ...
The second terms of a g.p 12 more than the first time give that the common ratio is half of the first terms find the third terms of the G.P
Why did the second term of the G.P want to be 12 more than the first term? Because it wanted to stand out from the crowd! But don't worry, I'll help you find the fourth term.
Let's assume the first term of the G.P is "a". According to the given information, the second term is "a + 12", and the common ratio is "a/2".
The formula to find the nth term of a G.P is: tn = a * r^(n-1), where "tn" represents the nth term, "a" is the first term, "r" is the common ratio, and "n" is the term number.
We want to find the fourth term, so let's substitute in the values:
t4 = a * (a/2)^(4-1)
Simplifying that, we get:
t4 = a * (a/2)^3
= a * (a^3/8)
= (a^4)/8
Therefore, the fourth term of the G.P is (a^4)/8.
To find the fourth term of the geometric progression (G.P), we need to first find the first term and the common ratio.
Let's assume the first term of the G.P is 'a', and the common ratio is 'r'.
According to the problem, the second term is 12 more than the first term:
So, the second term = first term + 12
=> ar = a + 12
It is also given that the common ratio is half of the first term:
So, r = a/2
Now we have two equations:
ar = a + 12 ...(Equation 1)
r = a/2 ...(Equation 2)
We can solve these equations simultaneously to find the values of 'a' and 'r'.
From Equation 2, we can substitute the value of r in Equation 1:
(a/2)a = a + 12
a^2/2 = a + 12
Multiply both sides by 2 to eliminate the fraction:
a^2 = 2a + 24
a^2 - 2a - 24 = 0
(a - 6)(a + 4) = 0
Setting each factor equal to zero:
a - 6 = 0 or a + 4 = 0
a = 6 or a = -4
Since a term of a geometric progression cannot be negative, we can consider a = 6 as the first term.
Now substituting the value of a in Equation 2:
r = 6/2
r = 3
So, the first term is 6 (a = 6) and the common ratio is 3 (r = 3).
To find the fourth term (T4) of the G.P, we can use the formula for finding the nth term of a G.P:
Tn = ar^(n-1)
Substituting the values, we have:
T4 = 6 * 3^(4 - 1)
T4 = 6 * 3^3
T4 = 6 * 27
T4 = 162
Therefore, the fourth term of the G.P is 162.
To find the fourth term of the geometric progression (G.P.), we need to first determine the first term and the common ratio.
Let's denote the first term as "a" and the common ratio as "r."
From the given information, we know that:
1) The second term of the G.P. is 12 more than the first term: a + 12.
2) The common ratio is half of the first term: r = a/2.
Now, we can set up the formula for the terms of the G.P.:
Second term: a * r = (a/2) * (a/2) = a^2 / 4 = a + 12.
Rearranging the equation:
a^2 = 4(a + 12),
a^2 = 4a + 48,
a^2 - 4a - 48 = 0.
To solve this quadratic equation, we can factor it:
(a - 8)(a + 6) = 0.
So, a = 8 or a = -6.
Since the terms of a geometric progression cannot be negative, we discard the negative value.
Hence, the first term (a) is 8.
To find the common ratio (r):
r = a/2 = 8/2 = 4.
Now, we can use the formula to find the fourth term (T4):
T4 = a * r^3 = 8 * 4^3 = 8 * 64 = 512.
Therefore, the fourth term of the geometric progression is 512.