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Haruka and Mustafa were asked to find an explicit formula for the sequence 4,12,36,108,dots, where the first term should be g(1). Haruka said the formula is g(n)=4*3^(n), and Mustafa said the formula is g(n)=4*4^(n-1). Which one of them is right? Choose 1 answer: (A) Only Haruka (B) Only Mustafa (c) Both Haruka and Mustafa (D) Neither Haruka nor Mustafa

Haruka and Mustafa were asked to find an explicit formula for the sequence \newline4,12,36,108,…4,12,36,108,\dots, where the first term should be \newlineg(1)g(1).\newlineHaruka said the formula is \newlineg(n)=4⋅3ng(n)=4\cdot3^{n}, and\newlineMustafa said the formula is \newlineg(n)=4⋅4n−1g(n)=4\cdot4^{n-1}.\newlineWhich one of them is right?\newlineChoose 11 answer:\newline(A) Only Haruka\newline(B) Only Mustafa\newline(C) Both Haruka and Mustafa\newline(D) Neither Haruka nor Mustafa

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Full solution

Q. Haruka and Mustafa were asked to find an explicit formula for the sequence \newline4,12,36,108,…4,12,36,108,\dots, where the first term should be \newlineg(1)g(1).\newlineHaruka said the formula is \newlineg(n)=4⋅3ng(n)=4\cdot3^{n}, and\newlineMustafa said the formula is \newlineg(n)=4⋅4n−1g(n)=4\cdot4^{n-1}.\newlineWhich one of them is right?\newlineChoose 11 answer:\newline(A) Only Haruka\newline(B) Only Mustafa\newline(C) Both Haruka and Mustafa\newline(D) Neither Haruka nor Mustafa
  1. Identifying the Pattern: We have the sequence: 4,12,36,108,…4, 12, 36, 108, \ldots\newlineTo determine which formula is correct, we need to identify the pattern of the sequence.
  2. Checking Haruka's Formula: The sequence appears to be geometric because each term is multiplied by the same factor to get the next term.\newlineTo find this common ratio rr, we divide the second term by the first term: r=124=3r = \frac{12}{4} = 3.
  3. Checking Mustafa's Formula: Now, let's check Haruka's formula: g(n)=4⋅3ng(n) = 4 \cdot 3^n.\newlineFor n=1n = 1, g(1)=4⋅31=4⋅3=12g(1) = 4 \cdot 3^1 = 4 \cdot 3 = 12, which does not match the first term of the sequence.\newlineTherefore, Haruka's formula is incorrect.
  4. Checking Mustafa's Formula: Now, let's check Haruka's formula: g(n)=4⋅3ng(n) = 4 \cdot 3^n. For n=1n = 1, g(1)=4⋅31=4⋅3=12g(1) = 4 \cdot 3^1 = 4 \cdot 3 = 12, which does not match the first term of the sequence. Therefore, Haruka's formula is incorrect.Let's check Mustafa's formula: g(n)=4⋅4(n−1)g(n) = 4 \cdot 4^{(n-1)}. For n=1n = 1, g(1)=4⋅4(1−1)=4⋅40=4⋅1=4g(1) = 4 \cdot 4^{(1-1)} = 4 \cdot 4^0 = 4 \cdot 1 = 4, which matches the first term of the sequence. For n=2n = 2, g(2)=4⋅4(2−1)=4⋅41=4⋅4=16g(2) = 4 \cdot 4^{(2-1)} = 4 \cdot 4^1 = 4 \cdot 4 = 16, which does not match the second term of the sequence. Therefore, Mustafa's formula is also incorrect.

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