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In a geometric sequence, the first term, a_(1), is equal to 3 , and the fifth term, a_(5), is equal to 48 . Which number represents the common ratio of the geometric sequence? r=1 r=2 r=3 r=4

In a geometric sequence, the first term, a1 a_{1} , is equal to 33 , and the fifth term, a5 a_{5} , is equal to 4848 . Which number represents the common ratio of the geometric sequence?\newliner=1 r=1 \newliner=2 r=2 \newliner=3 r=3 \newliner=4 r=4

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Q. In a geometric sequence, the first term, a1 a_{1} , is equal to 33 , and the fifth term, a5 a_{5} , is equal to 4848 . Which number represents the common ratio of the geometric sequence?\newliner=1 r=1 \newliner=2 r=2 \newliner=3 r=3 \newliner=4 r=4
  1. Identify Formula for nth Term: Identify the formula for the nth term of a geometric sequence. The nth term of a geometric sequence can be found using the formula an=a1×r(n−1)a_n = a_1 \times r^{(n-1)}, where ana_n is the nth term, a1a_1 is the first term, and rr is the common ratio.
  2. Set Up Equation: Set up the equation using the given terms.\newlineWe know that a1=3a_1 = 3 and a5=48a_5 = 48. We can use the formula from Step 11 to write the equation for the fifth term: 48=3⋅r5−148 = 3 \cdot r^{5-1}.
  3. Simplify Equation: Simplify the equation.\newline48=3×r448 = 3 \times r^4\newlineNow, we need to solve for rr.
  4. Divide to Isolate r4r^4: Divide both sides of the equation by 33 to isolate r4r^4.\newline483=r4\frac{48}{3} = r^4\newline16=r416 = r^4
  5. Take Fourth Root to Solve: Take the fourth root of both sides to solve for rr.r=161/4r = 16^{1/4}r=2r = 2

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