The 11th term of the geometric progression is 6144 or -6144.
Here's how we arrive at that answer, based on the information provided in the reference:
To find the 11th term, we first need to determine the common ratio (r) and the first term (a) of the geometric progression.
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Understanding Geometric Progression: A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The general term of a geometric progression can be written as arn-1 where 'a' is the first term, 'r' is the common ratio, and 'n' is the term number.
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Given Information:
- The second term is 12. Therefore, ar = 12.
- The sixth term is 192. Therefore, ar5 = 192.
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Finding the Common Ratio (r):
- Divide the equation for the sixth term by the equation for the second term: (ar5) / (ar) = 192 / 12
- This simplifies to r4 = 16.
- Taking the fourth root, we find r can be 2 or -2.
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Finding the First Term (a) for r=2:
- Using the second term (ar = 12) and r = 2: a * 2 = 12.
- Therefore, a = 6.
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Finding the First Term (a) for r=-2:
- Using the second term (ar = 12) and r = -2: a * -2 = 12
- Therefore, a = -6
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Calculating the 11th Term:
- The formula for the 11th term is ar10.
- For r=2 and a=6: The 11th term = 6 (2)10 = 6 1024 = 6144.
- For r=-2 and a=-6: The 11th term = -6 (-2)10= -6 1024 = -6144
Therefore, the 11th term of the geometric sequence is either 6144 or -6144 depending on the common ratio.
