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Divya and Miguel were asked to find an explicit formula for the sequence 80,40,20,10,dots. where the first term should be h(1). Divya said the formula is h(n)=80*((1)/(2))^(n-1), and Miguel said the formula is h(n)=160*((1)/(2))^(n)". " Which one of them is right? Choose 1 answer: (A) Only Divya (B) Only Miguel (c) Both Divya and Miguel (D) Neither Divya nor Miguel

Divya and Miguel were asked to find an explicit formula for the sequence 80,40,20,10,… 80,40,20,10, \ldots , where the first term should be h(1) h(1) .\newlineDivya said the formula is h(n)=80⋅(12)n−1 h(n)=80 \cdot\left(\frac{1}{2}\right)^{n-1} , and\newlineMiguel said the formula is h(n)=160⋅(12)n h(n)=160 \cdot\left(\frac{1}{2}\right)^{n} .\newlineWhich one of them is right?\newlineChoose 11 answer:\newline(A) Only Divya\newline(B) Only Miguel\newline(C) Both Divya and Miguel\newline(D) Neither Divya nor Miguel

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Q. Divya and Miguel were asked to find an explicit formula for the sequence 80,40,20,10,… 80,40,20,10, \ldots , where the first term should be h(1) h(1) .\newlineDivya said the formula is h(n)=80⋅(12)n−1 h(n)=80 \cdot\left(\frac{1}{2}\right)^{n-1} , and\newlineMiguel said the formula is h(n)=160⋅(12)n h(n)=160 \cdot\left(\frac{1}{2}\right)^{n} .\newlineWhich one of them is right?\newlineChoose 11 answer:\newline(A) Only Divya\newline(B) Only Miguel\newline(C) Both Divya and Miguel\newline(D) Neither Divya nor Miguel
  1. Question Prompt: The question prompt is: ＂Which formula correctly represents the explicit formula for the sequence 80,40,20,10,…80, 40, 20, 10, \ldots?＂
  2. Sequence Analysis: We have the sequence: 80,40,20,10,…80, 40, 20, 10, \ldots\newlineThis sequence is geometric because each term is obtained by multiplying the previous term by a common ratio.
  3. Finding the Common Ratio: To find the common ratio rr, we divide the second term by the first term: r=4080=12r = \frac{40}{80} = \frac{1}{2}.
  4. First Term of the Sequence: The first term of the sequence is 8080, so h(1)=80h(1) = 80.
  5. Explicit Formula for the nth Term: Using the common ratio and the first term, we can write the explicit formula for the nth term of the sequence as h(n)=h(1)⋅r(n−1)h(n) = h(1) \cdot r^{(n - 1)}.\newlineSubstituting the values we have h(n)=80⋅(12)(n−1)h(n) = 80 \cdot \left(\frac{1}{2}\right)^{(n - 1)}.
  6. Checking Divya's Formula: Now let's check Divya's formula: h(n)=80×(12)n−1h(n) = 80 \times \left(\frac{1}{2}\right)^{n - 1}.\newlineThis matches the formula we derived, so Divya's formula is correct.
  7. Checking Miguel's Formula: Let's check Miguel's formula: h(n)=160×(12)nh(n) = 160 \times \left(\frac{1}{2}\right)^n.\newlineTo see if this formula is correct, we can test it for n=1n = 1. If Miguel's formula is correct, h(1)h(1) should equal 8080.\newlineCalculating h(1)h(1) using Miguel's formula gives us h(1)=160×(12)1=160×12=80h(1) = 160 \times \left(\frac{1}{2}\right)^1 = 160 \times \frac{1}{2} = 80.\newlineThis is also correct because it gives us the first term of the sequence.
  8. Validity of Formulas: Since both Divya's and Miguel's formulas give the correct first term and follow the pattern of the sequence, both formulas are correct.

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