Question

The first term of a geometric sequence is 15, and the 5th term of the sequence is 3%7D%7B125%7D" id="TexFormula1" title="\frac{243}{125}" alt="\frac{243}{125}" align="absmiddle" class="latex-formula">
What are the geometric means between these terms?

Enter your answer by filling in the boxes to correctly complete the geometric sequence. Enter any fractions as simplified fractions.

15, __, __, __, $\frac{243}{125}$

Answer 1

The geometric sequence is $15,9,\frac{27}{5},\frac{81}{25}, \frac{243}{125}$

Explanation:

Given that the first term of the geometric sequence is 15

The fifth term of the sequence is $\frac{243}{125}$

We need to find the 2nd, 3rd and 4th term of the geometric sequence.

To find these terms, we need to know the common difference.

The common difference can be determined using the formula,

$a_n=a_1(r)^{n-1}$

where $a_1=15$ and $a_5=\frac{243}{125}$

For $n=5$, we have,

$\frac{243}{125}=15(r)^4$

Simplifying, we have,

$r=\frac{3}{5}$

Thus, the common difference is $r=\frac{3}{5}$

Now, we shall find the 2nd, 3rd and 4th terms by substituting $n=2,3,4$ in the formula $a_n=a_1(r)^{n-1}$

For $n=2$

$a_2=15(\frac{3}{5} )^{1}$

   $=9$  

Thus, the 2nd term of the sequence is 9

For $n=3$ , we have,

$a_3=15(\frac{3}{5} )^{2}$

   $=15(\frac{9}{25} )$

   $=\frac{27}{5}$

Thus, the 3rd term of the sequence is $\frac{27}{5}$

For $n=4$ , we have,

$a_4=15(\frac{3}{5} )^{3}$

    $=15(\frac{27}{25} )$

    $=\frac{81}{25}$

Thus, the 4th term of the sequence is $\frac{81}{25}$

Therefore, the geometric sequence is $15,9,\frac{27}{5},\frac{81}{25}, \frac{243}{125}$