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

a_(1)=80" and "a_(n)=-(1)/(4)a_(n-1)
Answer:
a_(n)=

Write an explicit formula that represents the sequence defined by the following recursive formula: $\newline$ $a_{1}=80 \text { and } a_{n}=-\frac{1}{4} a_{n-1}$ $\newline$ Answer: $a_{n}=$

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Evaluate an exponential function

Full solution

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

\newline

a_{1}=80 \text { and } a_{n}=-\frac{1}{4} a_{n-1}

\newline

Answer:

a_{n}=

Identify Pattern: To find the explicit formula for the sequence, we start by looking at the first few terms to identify a pattern. We know the first term is $a_{1} = 80$ .
Find Second Term: Now we apply the recursive formula to find the second term: $a_{2} = -\frac{1}{4}a_{1} = -\frac{1}{4} \times 80 = -20$ .
Find Third Term: Next, we find the third term using the recursive formula: $a_{3} = -\left(\frac{1}{4}\right)a_{2} = -\left(\frac{1}{4}\right) \times (-20) = 5$ .
Find Fourth Term: We continue this process to find the fourth term: $a_{4} = -\frac{1}{4}a_{3} = -\frac{1}{4} \times 5 = -1.25$ .
Determine Geometric Sequence: From these calculations, we can see that each term is $-\frac{1}{4}$ times the previous term. This is a geometric sequence with the first term $a_{1} = 80$ and a common ratio $r = -\frac{1}{4}$ .
Apply Explicit Formula: The explicit formula for a geometric sequence is given by $a_{n} = a_{1} \times r^{(n-1)}$ . Substituting the values we have, we get $a_{n} = 80 \times (-\frac{1}{4})^{(n-1)}$ .

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