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In a geometric sequence, the first term, 
a_(1), is equal to 2 , and the fourth term, 
a_(4), is equal to 128 . 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 22 , and the fourth term, a4 a_{4} , is equal to 128128 . 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 22 , and the fourth term, a4 a_{4} , is equal to 128128 . 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. We are given the first term a1a_{1} as 22 and the fourth term a4a_{4} as 128128. We need to find the common ratio rr.
  2. Use nth Term Formula: Use the formula for the nth term of a geometric sequence to express the fourth term.\newlineThe nth term of a geometric sequence is given by an=a1rn1a_n = a_1 \cdot r^{n-1}. For the fourth term, the formula is a4=a1r41=a1r3a_4 = a_1 \cdot r^{4-1} = a_1 \cdot r^3.
  3. Substitute Known Values: Substitute the known values into the formula.\newlineWe know that a1=2a_{1} = 2 and a4=128a_{4} = 128. So, we have 128=2×r3.128 = 2 \times r^{3}.
  4. Solve for Common Ratio: Solve for the common ratio rr. Divide both sides of the equation by 22 to isolate r3r^3. 128/2=r3128 / 2 = r^3 64=r364 = r^3
  5. Find Cube Root: Find the cube root of 6464 to solve for rr.\newlineThe cube root of 6464 is 44, so r=4r = 4.

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