To determine which one of them is correct, we need to evaluate both formulas with \( n = 1 \) and see which one produces the first term of the sequence, which is \( 100 \).
Rakesh's formula: \[ f(n) = 100 \cdot \left(\frac{1}{2}\right)^{n-1} \] Calculating \( f(1) \): \[ f(1) = 100 \cdot \left(\frac{1}{2}\right)^{1-1} = 100 \cdot \left(\frac{1}{2}\right)^0 = 100 \cdot 1 = 100 \] So, Rakesh's formula gives the first term \( 100 \).
Tessa's formula: \[ f(n) = 200 \cdot \left(\frac{1}{2}\right)^n \] Calculating \( f(1) \): \[ f(1) = 200 \cdot \left(\frac{1}{2}\right)^1 = 200 \cdot \frac{1}{2} = 100 \] So, Tessa's formula also gives the first term \( 100 \).
Now, let's evaluate further terms to ensure they match the sequence given:
Rakesh's formula for \( n \):
- \( f(2) = 100 \cdot \left(\frac{1}{2}\right)^{2-1} = 100 \cdot \frac{1}{2} = 50 \)
- \( f(3) = 100 \cdot \left(\frac{1}{2}\right)^{3-1} = 100 \cdot \left(\frac{1}{2}\right)^2 = 100 \cdot \frac{1}{4} = 25 \)
- \( f(4) = 100 \cdot \left(\frac{1}{2}\right)^{4-1} = 100 \cdot \left(\frac{1}{2}\right)^3 = 100 \cdot \frac{1}{8} = 12.5 \)
Tessa's formula for \( n \):
- \( f(2) = 200 \cdot \left(\frac{1}{2}\right)^2 = 200 \cdot \frac{1}{4} = 50 \)
- \( f(3) = 200 \cdot \left(\frac{1}{2}\right)^3 = 200 \cdot \frac{1}{8} = 25 \)
- \( f(4) = 200 \cdot \left(\frac{1}{2}\right)^4 = 200 \cdot \frac{1}{16} = 12.5 \)
Both formulas match the sequence \( 100, 50, 25, 12.5, \ldots \) correctly.
Thus, both Rakesh and Tessa provided correct formulas for the sequence.
Therefore, the correct answer is: (Choice C) Both Rakesh and Tessa
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