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Mohamed and Li Jing were asked to find an explicit formula for the sequence -5,-25,-125,-625,dots, where the first term should be g(1). Mohamed said the formula is g(n)=-5*5^(n), and Li Jing said the formula is g(n)=-5*5^(n-1). Which one of them is right? Choose 1 answer: (A) Only Mohamed (B) Only Li Jing (C) Both Mohamed and Li Jing (D) Neither Mohamed nor Li Jing

Mohamed and Li Jing were asked to find an explicit formula for the sequence \newline−5,−25,−125,−625,…-5,-25,-125,-625,\dots, where the first term should be \newlineg(1)g(1).\newlineMohamed said the formula is \newlineg(n)=−5⋅5ng(n)=-5\cdot5^{n}, and\newlineLi Jing said the formula is \newlineg(n)=−5⋅5n−1g(n)=-5\cdot5^{n-1}.\newlineWhich one of them is right?\newlineChoose 11 answer:\newline(A) Only Mohamed\newline(B) Only Li Jing\newline(C) Both Mohamed and Li Jing\newline(D) Neither Mohamed nor Li Jing

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Q. Mohamed and Li Jing were asked to find an explicit formula for the sequence \newline−5,−25,−125,−625,…-5,-25,-125,-625,\dots, where the first term should be \newlineg(1)g(1).\newlineMohamed said the formula is \newlineg(n)=−5⋅5ng(n)=-5\cdot5^{n}, and\newlineLi Jing said the formula is \newlineg(n)=−5⋅5n−1g(n)=-5\cdot5^{n-1}.\newlineWhich one of them is right?\newlineChoose 11 answer:\newline(A) Only Mohamed\newline(B) Only Li Jing\newline(C) Both Mohamed and Li Jing\newline(D) Neither Mohamed nor Li Jing
  1. Identify the pattern: We have the sequence: −5,−25,−125,−625,…-5, -25, -125, -625, \ldots\newlineIdentify the pattern in the sequence to determine if it is arithmetic or geometric.\newline−5-5 to −25-25 is multiplied by 55, −25-25 to −125-125 is multiplied by 55, and so on.\newlineThe sequence is geometric because each term is multiplied by a common ratio.
  2. Determine first term and common ratio: Determine the first term g(1)g(1) and the common ratio rr of the sequence.\newlineFirst term: g(1)=−5g(1) = -5\newlineCommon ratio: r=−25−5=5r = \frac{-25}{-5} = 5
  3. Check Mohamed's formula: Now, let's check Mohamed's formula: g(n)=−5⋅5ng(n) = -5 \cdot 5^n.\newlineIf n=1n = 1, g(1)g(1) should be −5-5.\newlineSubstitute n=1n = 1 into Mohamed's formula: g(1)=−5⋅51=−5⋅5=−25g(1) = -5 \cdot 5^1 = -5 \cdot 5 = -25.\newlineThis does not match the first term of the sequence, which is −5-5.
  4. Check Li Jing's formula: Now, let's check Li Jing's formula: g(n)=−5⋅5(n−1)g(n) = -5 \cdot 5^{(n-1)}.\newlineIf n=1n = 1, g(1)g(1) should be −5-5.\newlineSubstitute n=1n = 1 into Li Jing's formula: g(1)=−5⋅5(1−1)=−5⋅50=−5⋅1=−5g(1) = -5 \cdot 5^{(1-1)} = -5 \cdot 5^0 = -5 \cdot 1 = -5.\newlineThis matches the first term of the sequence.
  5. Conclusion: Since Li Jing's formula gives the correct first term and follows the geometric pattern of the sequence, Li Jing's formula is correct.\newlineMohamed's formula does not give the correct first term, so it is incorrect.

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