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In a geometric sequence, the first term,
a_(1), is equal to 8 , and the third term,
a_(3), is equal to 200 . Which number represents the common ratio of the geometric sequence?

r=3

r=4

r=5

r=6

In a geometric sequence, the first term, $a_{1}$ , is equal to $8$ , and the third term, $a_{3}$ , is equal to $200$ . Which number represents the common ratio of the geometric sequence? $\newline$ $r=3$ $\newline$ $r=4$ $\newline$ $r=5$ $\newline$ $r=6$

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Find the sum of a finite geometric series

Full solution

Q. In a geometric sequence, the first term,

a_{1}

, is equal to

8

, and the third term,

a_{3}

, is equal to

200

. Which number represents the common ratio of the geometric sequence?

\newline

r=3

\newline

r=4

\newline

r=5

\newline

r=6

Identify Given Terms: Identify the given terms in the geometric sequence. $\newline$ We are given the first term $a_{1} = 8$ and the third term $a_{3} = 200$ . In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio $(r)$ .
Write Third Term Formula: Write the formula for the third term of a geometric sequence. $\newline$ The third term $a_{3}$ can be found by multiplying the first term $a_{1}$ by the common ratio $(r)$ twice, since $a_{3} = a_{1} \times r^2$ .
Substitute Known Values: Substitute the known values into the formula. $\newline$ We have $a_{3} = 200$ and $a_{1} = 8$ , so we can write $200 = 8 \times r^{2}$ .
Solve for Common Ratio: Solve for the common ratio $r$ . Divide both sides of the equation by $8$ to isolate $r^2$ . $\frac{200}{8} = r^2$ $25 = r^2$
Find Value of r: Find the value of r by taking the square root of both sides. $\newline$ Since $r^2 = 25$ , we take the square root of $25$ to find $r$ . $\newline$ $r = \sqrt{25}$ $\newline$ $r = 5$

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