Divya and Miguel were asked to find an explicit formula for the sequence $80,40,20,10, \ldots$ , where the first term should be $h(1)$ . $\newline$ Divya said the formula is $h(n)=80 \cdot\left(\frac{1}{2}\right)^{n-1}$ , and $\newline$ Miguel said the formula is $h(n)=160 \cdot\left(\frac{1}{2}\right)^{n}$ . $\newline$ Which one of them is right? $\newline$ Choose $1$ 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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Math Problems

Algebra 1

Write variable expressions for geometric sequences

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Q. Divya and Miguel were asked to find an explicit formula for the sequence

80,40,20,10, \ldots

, where the first term should be

h(1)

\newline

Divya said the formula is

h(n)=80 \cdot\left(\frac{1}{2}\right)^{n-1}

, and

\newline

Miguel said the formula is

h(n)=160 \cdot\left(\frac{1}{2}\right)^{n}

\newline

Which one of them is right?

\newline

Choose

1

answer:

\newline

(A) Only Divya

\newline

(B) Only Miguel

\newline

\newline

(D) Neither Divya nor Miguel

Question Prompt: The question prompt is: ＂Which formula correctly represents the explicit formula for the sequence $80, 40, 20, 10, \ldots$ ?＂
Sequence Analysis: We have the sequence: $80, 40, 20, 10, \ldots$ $\newline$ This sequence is geometric because each term is obtained by multiplying the previous term by a common ratio.
Finding the Common Ratio: To find the common ratio $r$ , we divide the second term by the first term: $r = \frac{40}{80} = \frac{1}{2}$ .
First Term of the Sequence: The first term of the sequence is $80$ , so $h(1) = 80$ .
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 $h(n) = h(1) \cdot r^{(n - 1)}$ . $\newline$ Substituting the values we have $h(n) = 80 \cdot \left(\frac{1}{2}\right)^{(n - 1)}$ .
Checking Divya's Formula: Now let's check Divya's formula: $h(n) = 80 \times \left(\frac{1}{2}\right)^{n - 1}$ . $\newline$ This matches the formula we derived, so Divya's formula is correct.
Checking Miguel's Formula: Let's check Miguel's formula: $h(n) = 160 \times \left(\frac{1}{2}\right)^n$ . $\newline$ To see if this formula is correct, we can test it for $n = 1$ . If Miguel's formula is correct, $h(1)$ should equal $80$ . $\newline$ Calculating $h(1)$ using Miguel's formula gives us $h(1) = 160 \times \left(\frac{1}{2}\right)^1 = 160 \times \frac{1}{2} = 80$ . $\newline$ This is also correct because it gives us the first term of the sequence.
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.