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Woo-Jin and Kiran were asked to find an explicit formula for the sequence 64,16,4,1,dots, where the first term should be f(1). Woo-Jin said the formula is f(n)=64*((1)/(4))^(n), and Kiran said the formula is f(n)=16*((1)/(4))^(n-1)". " Which one of them is right? Choose 1 answer: (A) Only Woo-Jin (B) Only Kiran (c) Both Woo-Jin and Kiran (D) Neither Woo-Jin nor Kiran

Woo-Jin and Kiran were asked to find an explicit formula for the sequence 64,16,4,1,… 64,16,4,1, \ldots , where the first term should be f(1) f(1) .\newlineWoo-Jin said the formula is f(n)=64⋅(14)n f(n)=64 \cdot\left(\frac{1}{4}\right)^{n} , and\newlineKiran said the formula is f(n)=16⋅(14)n−1. f(n)=16 \cdot\left(\frac{1}{4}\right)^{n-1} . \newlineWhich one of them is right?\newlineChoose 11 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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Q. Woo-Jin and Kiran were asked to find an explicit formula for the sequence 64,16,4,1,… 64,16,4,1, \ldots , where the first term should be f(1) f(1) .\newlineWoo-Jin said the formula is f(n)=64⋅(14)n f(n)=64 \cdot\left(\frac{1}{4}\right)^{n} , and\newlineKiran said the formula is f(n)=16⋅(14)n−1. f(n)=16 \cdot\left(\frac{1}{4}\right)^{n-1} . \newlineWhich one of them is right?\newlineChoose 11 answer:\newline(A) Only Woo-Jin\newline(B) Only Kiran\newline(C) Both Woo-Jin and Kiran\newline(D) Neither Woo-Jin nor Kiran
  1. Question Prompt: The question prompt is: ＂Which formula correctly represents the sequence 64,16,4,1,…64, 16, 4, 1, \ldots?＂
  2. Sequence Analysis: We have the sequence: 64,16,4,1,…64, 16, 4, 1, \ldots\newlineThis sequence is geometric because each term is obtained by multiplying the previous term by a common ratio.
  3. Determine First Term and Common Ratio: Determine the first term f(1)f(1) and the common ratio rr of the sequence.\newlineFirst term: f(1)=64f(1) = 64\newlineTo find the common ratio, divide the second term by the first term: r=1664=14r = \frac{16}{64} = \frac{1}{4}
  4. Check Woo-Jin's Formula: Now, let's check Woo-Jin's formula: f(n)=64×(14)nf(n) = 64 \times (\frac{1}{4})^n\newlineIf we substitute n=1n = 1, we get f(1)=64×(14)1=64×14=16f(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.
  5. Check Kiran's Formula: Next, let's check Kiran's formula: f(n)=16×(14)(n−1)f(n) = 16 \times (\frac{1}{4})^{(n-1)} If we substitute n=1n = 1, we get f(1)=16×(14)(1−1)=16×(14)0=16×1=16f(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.
  6. Find Correct Formula: Since both Woo-Jin and Kiran's formulas do not yield the correct first term when n=1n = 1, we need to find the correct formula. The correct formula should give us 6464 when n=1n = 1.\newlineThe correct formula is f(n)=64×(1/4)(n−1)f(n) = 64 \times (1/4)^{(n-1)}, which gives us f(1)=64×(1/4)(1−1)=64×(1/4)0=64×1=64f(1) = 64 \times (1/4)^{(1-1)} = 64 \times (1/4)^0 = 64 \times 1 = 64.

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