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Write an explicit formula that represents the sequence defined by the following recursive formula:

a_(1)=100" and "a_(n)=-(1)/(5)a_(n-1)
Answer:
a_(n)=

Write an explicit formula that represents the sequence defined by the following recursive formula: $\newline$ $a_{1}=100 \text { and } a_{n}=-\frac{1}{5} a_{n-1}$ $\newline$ Answer: $a_{n}=$

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Evaluate exponential functions

Full solution

Q. Write an explicit formula that represents the sequence defined by the following recursive formula:

\newline

a_{1}=100 \text { and } a_{n}=-\frac{1}{5} a_{n-1}

\newline

Answer:

a_{n}=

Identify Pattern: To find an explicit formula for the sequence, we start by looking at the first few terms to identify a pattern. $\newline$ Given $a_{1}=100$ , we can find $a_{2}$ using the recursive formula $a_{n}=-\frac{1}{5}a_{n-1}$ . $\newline$ $a_{2} = -\frac{1}{5}a_{1} = -\frac{1}{5} \times 100 = -20$
Calculate $a_{2}$ : Next, we find $a_{3}$ using the same recursive formula. $a_{3} = -\frac{1}{5}a_{2} = -\frac{1}{5} \times (-20) = 4$
Calculate $a_{3}$ : We continue this process to find $a_{4}$ . $\newline$ $a_{4} = -\frac{1}{5}a_{3} = -\frac{1}{5} \times 4 = -0.8$
Find $a_{4}$ : From the pattern, we can see that each term is $-\frac{1}{5}$ times the previous term. This is a geometric sequence with the first term $a_{1} = 100$ and common ratio $r = -\frac{1}{5}$ . The explicit formula for a geometric sequence is $a_{n} = a_{1} \cdot r^{(n-1)}$ . Substituting the values we have, we get $a_{n} = 100 \cdot (-\frac{1}{5})^{(n-1)}$ .

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