Question

the product of 5 geometric means between 2 and 8 ​

Answer 1

Answer:

THE NUMBERS BETWEEN 2 AND 8 IN GP ARE:-

3,4,5,6 AND 7...

Step-by-step explanation:

NOW THE PRODUCT OF THESE NUMBERS WOULD BE 3×4×5×6×7 EQUALS 2520....

HOPE U LIKE IT,

PLEASE MARK ME BRAINLEIST

Answer 2

$\fontsize{18}{10}{\textup{\textbf{Five geometric means between 2 and 8 are}}}$

$2^\frac{4}{3},~~2^\frac{5}{3},~~2^2,~~2^\frac{7}{3},~~2^\frac{8}{3}$   or   $-2^\frac{4}{3},~~2^\frac{5}{3},~~-2^2,~~2^\frac{7}{3},~~-2^\frac{8}{3}.$

Step-by-step explanation:

Let a be the first term and r be the common ratio of the geometric progression

Then, first seven terms of the G.P. will be

$a,~~ar,~~ar^2,~~ar^3,~~ar^4,~~ar^5,~~ar^6.$

According to the given information, we must have

$a=2,\\\\ar^6=8.$

So,

$\dfrac{ar^6}{a}=\dfrac{8}{2}\\\\\Rightarrow r^6=4\\\\\Rightarrow r^3=\pm 2\\\\\Rightarrow r=(\pm2)^\frac{1}{3}.$

Therefore, the five geometric means between 2 and 8 are

$2\times2^\frac{1}{3},~~2\times2^\frac{2}{3},~~2\times2^\frac{3}{3},~~2\times2^\frac{4}{3},~~2\times2^\frac{5}{3}\\\\i.e.,~2^\frac{4}{3},~~2^\frac{5}{3},~~2^2,~~2^\frac{7}{3},~~2^\frac{8}{3}$

or

$2\times(-2)^\frac{1}{3},~~2\times(-2)^\frac{2}{3},~~2\times(-2)^\frac{3}{3},~~2\times(-2)^\frac{4}{3},~~2\times(-2)^\frac{5}{3}\\\\i.e.,~-2^\frac{4}{3},~~2^\frac{5}{3},~~-2^2,~~2^\frac{7}{3},~~-2^\frac{8}{3}.$

Thus, the five geometric means between 2 and 8 are

$2^\frac{4}{3},~~2^\frac{5}{3},~~2^2,~~2^\frac{7}{3},~~2^\frac{8}{3}$   or   $-2^\frac{4}{3},~~2^\frac{5}{3},~~-2^2,~~2^\frac{7}{3},~~-2^\frac{8}{3}.$