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In a geometric sequence, the first term, 
a_(1), is equal to 1 , and the third term, 
a_(3), is equal to 9 . 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 11 , and the third term, a3 a_{3} , is equal to 99 . 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 11 , and the third term, a3 a_{3} , is equal to 99 . 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 Sequence Properties: 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 rr is the common ratio.
  2. Set Up Common Ratio Equation: Set up the equation to find the common ratio using the given terms.\newlineWe know that a1=1a_{1} = 1 and a3=9a_{3} = 9. Using the formula for the nth term of a geometric sequence, we can write the equation for the third term as follows:\newlinea3=a1×r31a_{3} = a_{1} \times r^{3-1}\newline9=1×r29 = 1 \times r^{2}
  3. Solve for Common Ratio: Solve for the common ratio rr. Now we simplify and solve for rr: 9=r29 = r^2 To find rr, we take the square root of both sides of the equation: r=9r = \sqrt{9} r=3r = 3

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