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In a geometric sequence, the first term, 
a_(1), is equal to 7 , and the fifth term, 
a_(5), is equal to 112 . 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 77 , and the fifth term, a5 a_{5} , is equal to 112112 . 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

Full solution

Q. In a geometric sequence, the first term, a1 a_{1} , is equal to 77 , and the fifth term, a5 a_{5} , is equal to 112112 . 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 Given Terms: Identify the given terms in the geometric sequence.\newlineWe are given the first term a1a_{1} as 77 and the fifth term a5a_{5} as 112112.
  2. Recall Formula: Recall the formula for the nnth term of a geometric sequence.\newlineThe nnth term of a geometric sequence is given by an=a1r(n1)a_{n} = a_{1} \cdot r^{(n-1)}, where rr is the common ratio.
  3. Set Up Equation: Set up the equation to find the common ratio using the given terms.\newlineWe have a5=a1×r51a_{5} = a_{1} \times r^{5-1}, which simplifies to 112=7×r4112 = 7 \times r^4.
  4. Solve for Ratio: Solve for the common ratio rr.\newlineDivide both sides of the equation by 77 to isolate r4r^4.\newline1127=r4\frac{112}{7} = r^4\newline16=r416 = r^4
  5. Find Fourth Root: Find the fourth root of 1616 to solve for rr. \newliner=1614r = 16^{\frac{1}{4}}\newliner=2r = 2

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