Woo-Jin and Kiran were asked to find an explicit formula for the sequence $64,16,4,1, \ldots$ , where the first term should be $f(1)$ . $\newline$ Woo-Jin said the formula is $f(n)=64 \cdot\left(\frac{1}{4}\right)^{n}$ , and $\newline$ Kiran said the formula is $f(n)=16 \cdot\left(\frac{1}{4}\right)^{n-1} .$ $\newline$ Which one of them is right? $\newline$ Choose $1$ answer: $\newline$ (A) Only Woo-Jin $\newline$ (B) Only Kiran $\newline$ (C) Both Woo-Jin and Kiran $\newline$ (D) Neither Woo-Jin nor Kiran

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Algebra 1

Write variable expressions for geometric sequences

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Q. Woo-Jin and Kiran were asked to find an explicit formula for the sequence

64,16,4,1, \ldots

, where the first term should be

f(1)

\newline

Woo-Jin said the formula is

f(n)=64 \cdot\left(\frac{1}{4}\right)^{n}

, and

\newline

Kiran said the formula is

f(n)=16 \cdot\left(\frac{1}{4}\right)^{n-1} .

\newline

Which one of them is right?

\newline

Choose

1

answer:

\newline

(A) Only Woo-Jin

\newline

(B) Only Kiran

\newline

\newline

(D) Neither Woo-Jin nor Kiran

Question Prompt: The question prompt is: ＂Which formula correctly represents the sequence $64, 16, 4, 1, \ldots$ ?＂
Sequence Analysis: We have the sequence: $64, 16, 4, 1, \ldots$ $\newline$ This sequence is geometric because each term is obtained by multiplying the previous term by a common ratio.
Determine First Term and Common Ratio: Determine the first term $f(1)$ and the common ratio $r$ of the sequence. $\newline$ First term: $f(1) = 64$ $\newline$ To find the common ratio, divide the second term by the first term: $r = \frac{16}{64} = \frac{1}{4}$
Check Woo-Jin's Formula: Now, let's check Woo-Jin's formula: $f(n) = 64 \times (\frac{1}{4})^n$ $\newline$ If we substitute $n = 1$ , we get $f(1) = 64 \times (\frac{1}{4})^1 = 64 \times \frac{1}{4} = 16$ , which is not the first term of the sequence. Therefore, Woo-Jin's formula is incorrect.
Check Kiran's Formula: Next, let's check Kiran's formula: $f(n) = 16 \times (\frac{1}{4})^{(n-1)}$ If we substitute $n = 1$ , we get $f(1) = 16 \times (\frac{1}{4})^{(1-1)} = 16 \times (\frac{1}{4})^0 = 16 \times 1 = 16$ , which is also not the first term of the sequence. Therefore, Kiran's formula is incorrect as well.
Find Correct Formula: Since both Woo-Jin and Kiran's formulas do not yield the correct first term when $n = 1$ , we need to find the correct formula. The correct formula should give us $64$ when $n = 1$ . $\newline$ The correct formula is $f(n) = 64 \times (1/4)^{(n-1)}$ , which gives us $f(1) = 64 \times (1/4)^{(1-1)} = 64 \times (1/4)^0 = 64 \times 1 = 64$ .