Question

Khaled and Wilma were asked to find an explicit formula for the sequence
\[1\,,\,3\,,\,9\,,\,27,\unicode{0x2026}\], where the first term should be
\[f(1)\].
Khaled said the formula is
\[f(n)=1\cdot3^{{n-1}}\], and
Wilma said the formula is
\[f(n)=1\cdot3^{{n}}\].
Which one of them is right?
Choose 1 answer:
Choose 1 answer:
(Choice A) Only Khaled
A
Only Khaled
(Choice B) Only Wilma
B
Only Wilma
(Choice C) Both Khaled and Wilma
C
Both Khaled and Wilma
(Choice D) Neither Khaled nor Wilma
D
Neither Khaled nor Wilma

Answer 1

To determine which formula correctly describes the sequence \(1\,,\,3\,,\,9\,,\,27,\ldots\), we will analyze both proposed formulas.

The given sequence can be expressed as follows:
- The first term \(f(1) = 1\)
- The second term \(f(2) = 3\)
- The third term \(f(3) = 9\)
- The fourth term \(f(4) = 27\)

We can observe that each term in the sequence can be expressed as \(3^{n-1}\):
- \(f(1) = 3^{1-1} = 3^0 = 1\)
- \(f(2) = 3^{2-1} = 3^1 = 3\)
- \(f(3) = 3^{3-1} = 3^2 = 9\)
- \(f(4) = 3^{4-1} = 3^3 = 27\)

Therefore, Khaled's formula \(f(n) = 1 \cdot 3^{n-1}\) is correct.

Now let's look at Wilma's formula \(f(n) = 1 \cdot 3^n\):
- \(f(1) = 3^1 = 3\) (which does not match since \(f(1)\) should be \(1\))
- \(f(2) = 3^2 = 9\) (which does not match since \(f(2)\) should be \(3\))
- \(f(3) = 3^3 = 27\) (which does not match since \(f(3)\) should be \(9\))
- \(f(4) = 3^4 = 81\) (which does not match since \(f(4)\) should be \(27\))

Wilma's formula is incorrect for every term in the sequence.

Thus, the correct answer is:

(Choice A) Only Khaled.