Question

Six numbers are in G.P such that their product is 512 iſ the fourth number is 4, then the second number is​

Answer 1

Given:

Six numbers are in G.P.

Product of the numbers is 512.

Fourth number is 4.

To find:

Second number = ?

Solution:

nth term in a GP is given by:

$a_n=ar^{n-1}$

Fourth number is 4.

i.e. $ar^3=4 ...... (1)$

The six numbers in GP can be:

$a, ar, ar^2, ar^3, ar^4, ar^5$

Taking Their product and putting equal to 512.

$a \times ar \times ar^2\times ar^3\times ar^4\times ar^5 = 512\\\Rightarrow a^6r^{15} = 512$

Rewriting above equation:

$\Rightarrow a\times a^5r^{15} = 512\\\Rightarrow a \times (ar^3)^5 = 512\\\\\text{Using equation (1), putting } ar^3 = 4\\\\\Rightarrow a \times 4^5 = 512\\\Rightarrow a = \dfrac{512}{1024}\\\Rightarrow a = \dfrac{1}{2}$

Putting value of a in equation (1):

$\dfrac{1}{2}r^3=4 \\\Rightarrow r^3 = 8\\\Rightarrow r = 2$

Now, 2nd term in a GP is given as:

$a_2 = ar\\\Rightarrow a_2 = \dfrac{1}{2}\times 2\\\Rightarrow a_2 = 1$

So, the second term is = 1