Three numbers form a GP. If the first and third numbers are 5 and 245 respectively, find two possible values…

Question

Three numbers form a GP. If the first and third numbers are 5 and 245 respectively, find two possible values for the middle number.

ScanSolve AI Tutor · Accepted Answer

Let the three numbers in the Geometric Progression (GP) be $a$ , $ar$ , and $ar^2$ , where $a$ is the first term and $r$ is the common ratio.

Identify the given values.
The first term is $a = 5$ .
The third term is $ar^2 = 245$ .
Substitute the value of $a$ into the equation for the third term to find the common ratio $r$ .
$5r^2 = 245$
Divide both sides by 5:
$r^2 = \frac{245}{5}$ $r^2 = 49$
Solve for $r$ .
Take the square root of both sides:
$r = \pm\sqrt{49}$ $r = \pm 7$
There are two possible values for the common ratio: $r=7$ and $r=-7$ .
Calculate the middle number ( $ar$ ) for each possible value of $r$ .
Case 1: If $r = 7$
The middle number is $ar = 5 \times 7 = 35$ .

Case 2: If $r = -7$
The middle number is $ar = 5 \times (-7) = -35$ .

The two possible values for the middle number are $35 \text{ and } -35$ .

Three numbers form a GP. If the first and third numbers are 5 and 245 respectively, find two possible values for the middle number.