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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 72 . Which number represents the common ratio of the geometric sequence?

r=2

r=3

r=4

r=5

In a geometric sequence, the first term, a1 a_{1} , is equal to 88 , and the third term, a3 a_{3} , is equal to 7272 . Which number represents the common ratio of the geometric sequence?\newliner=2 r=2 \newliner=3 r=3 \newliner=4 r=4 \newliner=5 r=5

Full solution

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 7272 . Which number represents the common ratio of the geometric sequence?\newliner=2 r=2 \newliner=3 r=3 \newliner=4 r=4 \newliner=5 r=5
  1. Understand Geometric Sequences: Understand the properties of a geometric sequence. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio rr. The nnth term of a geometric sequence can be found using the formula an=a1×r(n1)a_{n} = a_{1} \times r^{(n-1)}, where a1a_{1} is the first term and ana_{n} is the nnth term.
  2. Set Up Equation for Common Ratio: Set up the equation to find the common ratio using the given terms.\newlineWe are given the first term a1=8a_1 = 8 and the third term a3=72a_3 = 72. We can use the formula for the nth term of a geometric sequence to write the equation for the third term: a3=a1r31=8r2a_3 = a_1 \cdot r^{3-1} = 8 \cdot r^2.
  3. Substitute Known Values: Substitute the known values into the equation. 72=8×r272 = 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. 728=r2\frac{72}{8} = r^2 9=r29 = r^2
  5. Find Value of r: Find the value of r by taking the square root of both sides.\newliner=9r = \sqrt{9}\newliner=3r = 3 or r=3r = -3
  6. Determine Appropriate rr: Determine the appropriate value of rr. Since we are looking for a common ratio in a geometric sequence, and the sequence is typically considered with positive terms, we take r=3r = 3 as the common ratio.

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