The second term of a GP is 12, more than the first term, given that the common ratio is half of the first. Find the 3rd term of the GP
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To find the third term of the geometric progression (GP), we need to determine the common ratio and the first term. Let's break down the given information:
1. The second term of the GP is 12 more than the first term.
2. The common ratio is half of the first term.
Let's denote:
- The first term as 'a',
- The second term as 'a + 12' (since it is 12 more than the first term),
- The common ratio as 'r'.
Since the common ratio is half of the first term, we can express it as:
r = 1/2 * a
Now, we can use this information to find the values of 'a' and 'r':
Substitute the value of 'r' in the expression of the second term:
a + 12 = r * a
Since r = 1/2 * a, substitute this value as well:
a + 12 = (1/2 * a) * a
Simplify equation further:
a + 12 = a^2/2
Now, rearrange the equation to solve for the first term 'a':
Multiply the equation by 2 to remove the fraction:
2 * (a + 12) = a^2
Expand and rearrange the equation:
2a + 24 = a^2
a^2 - 2a - 24 = 0
This equation is quadratic. We can solve it by factoring or by using the quadratic formula. Factoring does not yield integer solutions, so let's use the quadratic formula:
a = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a^2 - 2a - 24 = 0, the coefficients are:
a = 1, b = -2, c = -24
Substitute these values into the quadratic formula:
a = (-(-2) ± √((-2)^2 - 4(1)(-24))) / (2(1))
a = (2 ± √(4 + 96)) / 2
a = (2 ± √100) / 2
a = (2 ± 10) / 2
Now, consider both possible roots:
1. When a = (2 + 10) / 2 = 12/2 = 6
2. When a = (2 - 10) / 2 = -8/2 = -4
Since the terms of the GP cannot be negative, we discard the second root (-4) and consider the first root (a = 6) as the first term of the GP.
Now that we have determined the first term 'a', we can find the common ratio 'r':
r = 1/2 * a
r = 1/2 * 6
r = 3
Now, we have the first term (a = 6) and the common ratio (r = 3) of the GP. We can use these values to find the third term:
The third term, T3, can be calculated using the formula for the nth term of a GP:
Tn = a * r^(n-1)
Plug in the values:
T3 = 6 * 3^(3-1)
T3 = 6 * 3^2
T3 = 6 * 9
T3 = 54
Therefore, the third term of the GP is 54.