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In a geometric sequence, the first term, a_(1), is equal to 8 , and the third term, a_(3), is equal to 200 . Which number represents the common ratio of the geometric sequence? r=3 r=4 r=5 r=6

In a geometric sequence, the first term, a1 a_{1} , is equal to 88 , and the third term, a3 a_{3} , is equal to 200200 . Which number represents the common ratio of the geometric sequence?\newliner=3 r=3 \newliner=4 r=4 \newliner=5 r=5 \newliner=6 r=6

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Q. In a geometric sequence, the first term, a1 a_{1} , is equal to 88 , and the third term, a3 a_{3} , is equal to 200200 . Which number represents the common ratio of the geometric sequence?\newliner=3 r=3 \newliner=4 r=4 \newliner=5 r=5 \newliner=6 r=6
  1. Identify Given Terms: Identify the given terms in the geometric sequence.\newlineWe are given the first term a1=8a_{1} = 8 and the third term a3=200a_{3} = 200. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio (r)(r).
  2. Write Third Term Formula: Write the formula for the third term of a geometric sequence.\newlineThe third term a3a_{3} can be found by multiplying the first term a1a_{1} by the common ratio (r)(r) twice, since a3=a1×r2a_{3} = a_{1} \times r^2.
  3. Substitute Known Values: Substitute the known values into the formula.\newlineWe have a3=200a_{3} = 200 and a1=8a_{1} = 8, so we can write 200=8×r2200 = 8 \times r^{2}.
  4. Solve for Common Ratio: Solve for the common ratio rr. Divide both sides of the equation by 88 to isolate r2r^2. 2008=r2\frac{200}{8} = r^2 25=r225 = r^2
  5. Find Value of r: Find the value of r by taking the square root of both sides.\newlineSince r2=25r^2 = 25, we take the square root of 2525 to find rr.\newliner=25r = \sqrt{25}\newliner=5r = 5

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