Question

Rakesh and Tessa were asked to find an explicit formula for the sequence
\[100\,,\,50\,,\,25\,,\,12.5,\unicode{0x2026}\], where the first term should be
\[f(1)\].
Rakesh said the formula is

\[f(n)=100\cdot\left(\dfrac{1}{2}\right)^{{n-1}}\], and
Tessa said the formula is

\[f(n)=200\cdot\left(\dfrac{1}{2}\right)^{{n}}\].
Which one of them is right?
Choose 1 answer:
Choose 1 answer:
(Choice A) Only Rakesh
A
Only Rakesh
(Choice B) Only Tessa
B
Only Tessa
(Choice C) Both Rakesh and Tessa
C
Both Rakesh and Tessa
(Choice D) Neither Rakesh nor Tessa
D
Neither Rakesh nor Tessa

Answer 1

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