The common ratio of the geometric progression is 2.
Here's how to find it:
Understanding Geometric Progressions
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 formula for the nth term (aₙ) of a geometric progression is:
*aₙ = a r^(n-1)**
Where:
- a is the first term.
- r is the common ratio.
- n is the term number.
Solving for the Common Ratio
We are given:
- The 6th term (a₆) = 192
- The 2nd term (a₂) = 12
From the formula above we have:
- a₂ = ar^(2-1) = ar = 12
- a₆ = ar^(6-1) = ar^5 = 192
We can express a₆ as a₂ times r⁴:
- a₆ = a₂ * r⁴
- 192 = 12 * r⁴
Now we solve for r⁴:
- r⁴ = 192 / 12
- r⁴ = 16
Finally, we solve for r:
- r = ⁴√16
- r = 2
Therefore, the common ratio (r) is 2, as stated in the reference which says "Thus 192 = 12*r⁴. r⁴ = 16. r=2."
