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Recidivist

To find the eighth term of a geometric sequence, we can use the formula for the nth term of a geometric sequence:

a_n = a_1 * r^(n-1)

where a_1 is the first term of the sequence, r is the common ratio, and n is the position of the term.

In this case, the first term is a_1 = 5/3 and the common ratio is r = (-1)/(3/5) = -5/3. Plugging these values into the formula, we get:

a_8 = (5/3) * (-5/3)^(8-1)

= (5/3) * (-5/3)^7

= (5/3) * (-78125/2187)

= (5/3) * (-35/9)

= (-25/27)

Therefore, the eighth term of the sequence is a8 = -25/27.

semsee45

Answer:

[tex]a_8=-\dfrac{729}{15625}[/tex]

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Geometric sequence}\\\\$a_n=ar^{n-1}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\\phantom{ww}$\bullet$ $r$ is the common ratio.\\\phantom{ww}$\bullet$ $a_n$ is the $n$th term.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]

Given geometric sequence:

[tex]\dfrac{5}{3},\;-1,\;\dfrac{3}{5},\;...[/tex]

To find the common ratio, divide a term by the previous term:

[tex]\implies r=\dfrac{a_3}{a_2}=\dfrac{\frac{3}{5}}{-1}=-\dfrac{3}{5}[/tex]

Substitute the found common ratio and given first term into the formula to create an equation for the nth term:

[tex]a_n=\dfrac{5}{3}\left(-\dfrac{3}{5}\right)^{n-1}[/tex]

To find the 8th term, substitute n = 8 into the equation:

[tex]\implies a_8=\dfrac{5}{3}\left(-\dfrac{3}{5}\right)^{8-1}[/tex]

[tex]\implies a_8=\dfrac{5}{3}\left(-\dfrac{3}{5}\right)^{7}[/tex]

[tex]\implies a_8=\dfrac{5}{3}\left(-\dfrac{2187}{78125}\right)[/tex]

[tex]\implies a_8=-\dfrac{729}{15625}[/tex]

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