Question

36, ___, ___, 32/3 is a geometric sequence, with some terms missing. First, please solve for "r", the common ratio between each term. What is the value of r? Please enter your answer as a fraction, such as 6/7 or 1/5. *

Answer 1

Answer:

$\frac{2}{3}$

Step-by-step explanation:

Given:

36, ___, ___, 32/3 is a geometric sequence

We need to find common ratio, r.

Solution:

As here is geometric sequence:-

First term = $ar$ = 36

Fourth term = $ar^{4}$ = $\frac{32}{3}$

$r=?$

We can write $ar^{4}$ as $ar\times r^{3}$

$( \ ar=36, \ given)$

$ar^{4} =\frac{32}{3} \ given$

$ar\times r^{3}=\frac{32}{3}$

$36\times r^{3} =\frac{32}{3}$

Dividing both sides by 36

$\frac{36}{36} \times r^{3} =\frac{32}{3\times36} \\\\ r^{3} =\frac{32}{108} \\Taking\ cube\ root\ both\ sides\\\sqrt[3]{r^{3} } =\sqrt[3]{\frac{32}{108} } \\ \\ r=\sqrt[3]{\frac{8}{27} } \\ \\ r=\sqrt[3]{\frac{2\times2\times2}{3\times3\times3} }\\ \\ r=\frac{2}{3}$

Thus, common ratio, r is $\frac{2}{3}$

2nd term = $a r^{2} =ar\times r=36\times\frac{2}{3} =24$

3rd term=$ar^{3} =ar\times r^{2} =36\times (\frac{2}{3} )^{2} =36\times=\frac{4}{9} =16$

Answer 2

Answer:

The geometric sequence is

36,24,16,$\frac{32}3$,.......

Step-by-step explanation:

The first term of the g.s is 36.

The 4th term of the given sequence is $\frac{32}{3}$.

The $n^{th}$ term of the geometric sequence is

$T_n=ar^{n-1}$

Where a is the first term of the geometric sequence and r is the geometric sequence.

Then 4th term of the sequence is

$T_4=ar^{4-1}$

$\Rightarrow \frac{32}3=36r^3$

$\Rightarrow r^3=\frac{32}{3\times 36}$

$\Rightarrow r=\sqrt[3]{ \frac{8}{27}}$

$\Rightarrow r=\frac{2}{3}$

Then, second term of the sequence   $=36\times( \frac 23)^{2-1}$

                                                              $=36\times \frac 23$

                                                             =24

The third term of the sequence is$=36\times( \frac 23)^{3-1}$

                                                       $=36\times( \frac 23)^2$

                                                        =16

The geometric sequence is

36,24,16,$\frac{32}3$,.......