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an=5(2)^n−1
A sub n is equal to 5 times 2 raised to the n minus 1 power

an=3(7)^n
A sub n is equal to 3 times 7 to the n th power

an=2(5)^n−1
A sub n is equal to 2 times 5 raised to the n minus 1 power

an=5(2)^n+1

5 answers

It looks like you provided a set of formulas for \( a_n \) along with their verbal representations. If you need help clarifying these formulas or if you have a specific question related to them, please provide more information so I can assist you further. Here are the verbal representations again for clarity:

  1. \( a_n = 5(2)^{n-1} \)

    • "A sub n is equal to 5 times 2 raised to the (n minus 1) power."

  2. \( a_n = 3(7)^n \)

    • "A sub n is equal to 3 times 7 to the n-th power."

  3. \( a_n = 2(5)^{n-1} \)

    • "A sub n is equal to 2 times 5 raised to the (n minus 1) power."

  4. \( a_n = 5(2)^{n+1} \)

    • "A sub n is equal to 5 times 2 raised to the (n plus 1) power."

If you have a question about these expressions or need assistance with a related topic, feel free to ask!

can you try and give me an answer?

It seems like you're asking for clarification or a specific answer related to the expressions you've provided. However, the context of your request is not clear.

If you want to compare or analyze these expressions, here are some observations:

  1. Exponential Growth Patterns:

    • \(a_n = 5(2)^{n-1}\) and \(a_n = 5(2)^{n+1}\) both involve base 2, which means they will grow exponentially based on the value of \(n\), but with different coefficients.
    • \(a_n = 3(7)^n\) involves base 7, which grows faster than base 2 as \(n\) increases.
    • \(a_n = 2(5)^{n-1}\) involves base 5, which also grows faster than base 2 but slower than base 7.

  2. Initial Values:

    • The sequences begin with different initial values depending on \(n\). For instance, for \(n=0\):
      • \(a_0 = 5(2)^{-1} = 2.5\)
      • \(a_0 = 3(7)^0 = 3\)
      • \(a_0 = 2(5)^{-1} = 0.4\)
      • \(a_0 = 5(2)^{1} = 10\)

  3. Finding an nth Value: If you need to find a specific value for \(n\) for one of the sequences, please specify the value of \(n\) you want analyzed or any other specific questions you have related to this topic.

If you could clarify what kind of answer you are seeking (e.g., a specific calculation, analysis, or comparison), I'll be more than happy to assist!

yes I can.

Convert the recursive form a1=5, an=2an−1 to the explicit form.

To convert the recursive definition \( a_1 = 5 \) and \( a_n = 2a_{n-1} \) into an explicit form, we can derive the formula step by step.

Steps to Derive the Explicit Formula:

  1. Write out the first few terms:

    • For \( n = 1 \): \[ a_1 = 5 \]

    • For \( n = 2 \): \[ a_2 = 2a_1 = 2 \cdot 5 = 10 \]

    • For \( n = 3 \): \[ a_3 = 2a_2 = 2 \cdot 10 = 20 \]

    • For \( n = 4 \): \[ a_4 = 2a_3 = 2 \cdot 20 = 40 \]

    • For \( n = 5 \): \[ a_5 = 2a_4 = 2 \cdot 40 = 80 \]

    From the results, we see the sequence: \( a_1 = 5, a_2 = 10, a_3 = 20, a_4 = 40, a_5 = 80 \).

  2. Identify a pattern: From the computed values, it looks like each term is 5 multiplied by \( 2^{n-1} \):

    • \( a_1 = 5 \cdot 2^{0} = 5 \)
    • \( a_2 = 5 \cdot 2^{1} = 10 \)
    • \( a_3 = 5 \cdot 2^{2} = 20 \)
    • \( a_4 = 5 \cdot 2^{3} = 40 \)
    • \( a_5 = 5 \cdot 2^{4} = 80 \)

  3. Explicit Formula: The pattern we deduced can be expressed in an explicit form: \[ a_n = 5 \cdot 2^{n-1} \]

Conclusion:

The explicit formula for the recursive sequence defined by \( a_1 = 5 \) and \( a_n = 2a_{n-1} \) is: \[ \boxed{a_n = 5 \cdot 2^{n-1}} \]