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In a geometric sequence, the first term, 
a_(1), is equal to 1 , and the fourth term, 
a_(4), is equal to 64 . 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 fourth term, a4 a_{4} , is equal to 6464 . 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 fourth term, a4 a_{4} , is equal to 6464 . 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. Identify Given Terms: Identify the given terms in the geometric sequence.\newlineWe are given the first term a1a_{1} and the fourth term a4a_{4} of the geometric sequence. The first term is 11 a1=1a_{1} = 1 and the fourth term is 6464 a4=64a_{4} = 64.
  2. Recall Formula: Recall the formula for the nnth term of a geometric sequence.\newlineThe nnth term of a geometric sequence can be found using the formula an=a1r(n1)a_n = a_1 \cdot r^{(n-1)}, where a1a_1 is the first term, rr is the common ratio, and nn is the term number.
  3. Set Up Equation: Set up the equation to find the common ratio using the given terms.\newlineWe know that a4=a1×r41a_{4} = a_{1} \times r^{4-1}, which simplifies to 64=1×r364 = 1 \times r^3.
  4. Solve for Ratio: Solve for the common ratio rr. Since a1=1a_{1} = 1, the equation simplifies to 64=r364 = r^3. To find rr, we need to take the cube root of both sides of the equation. The cube root of 6464 is 44, so r=4r = 4.

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