Question
Write the explicit formula for the geometric sequence represented by the recursive formula. \( a_{1}=4 \) \( a_{n}=5 \cdot a_{n-1} \) \( a_{n}=4(5)^{n-1} \) \( a_{n}=5(4)^{n} \) \( a_{n}=4(5)^{n} \) \( a_{n}=5(4)^{n-1} \)
Ask by Lynch Donnelly. in the United States
Feb 28,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The explicit formula for the geometric sequence is \( a_{n} = 4 \times 5^{n-1} \).
Solution
The recursion is given by:
a₁ = 4
aₙ = 5 · aₙ₋₁
Since this is a geometric sequence with common ratio r = 5, the explicit formula for a geometric sequence is:
aₙ = a₁ · r^(n - 1)
Plugging in the values:
aₙ = 4 · 5^(n - 1)
So the correct explicit formula is:
aₙ = 4(5)^(n - 1)
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Simplify this solution Beyond the Answer
The recursive formula you've provided defines a geometric sequence where the first term \( a_1 = 4 \) and each term is multiplied by a common ratio of \( 5 \). To derive the explicit formula, we start with the first term and use the common ratio. Thus, the correct explicit formula is \( a_n = 4(5)^{n-1} \), where \( n \) denotes the position in the sequence. Fun fact: Geometric sequences are all about multiplication and the 'speed' of growth! While linear sequences add a fixed amount each time, geometric sequences can skyrocket as they build on themselves with the common ratio. In this case, after just a few terms, you’ll see how quickly the values escalate!
