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${[a(1)=(5)/(3)],[a(n)=a(n-1)*(-9)]:} What is the 3^("rd ") term in the sequence?$

$\left\{\begin{array}{l} a(1)=\frac{5}{3} \\ a(n)=a(n-1) \cdot(-9) \end{array}\right.$ $\newline$ What is the $3^{\text {rd }}$ term in the sequence?

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Geometric sequences

Full solution

Q.

\left\{\begin{array}{l} a(1)=\frac{5}{3} \\ a(n)=a(n-1) \cdot(-9) \end{array}\right.

\newline

What is the

3^{\text {rd }}

term in the sequence?

Identify first term and common ratio: Identify the first term of the sequence and the common ratio. $\newline$ The first term $a(1)$ is given as $\frac{5}{3}$ . The sequence is defined recursively, with each term being $-9$ times the previous term. Therefore, the common ratio is $-9$ .
Calculate second term: Calculate the second term using the first term and the common ratio. $\newline$ The second term $a(2)$ is $a(1)$ multiplied by the common ratio. $\newline$ $a(2) = a(1) \times (-9)$ $\newline$ $a(2) = \left(\frac{5}{3}\right) \times (-9)$ $\newline$ $a(2) = -15$
Calculate third term: Calculate the third term using the second term and the common ratio. $\newline$ The third term $a(3)$ is $a(2)$ multiplied by the common ratio. $\newline$ $a(3) = a(2) \times (-9)$ $\newline$ $a(3) = -15 \times (-9)$ $\newline$ $a(3) = 135$

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