Khaled and Wilma were asked to find an explicit formula for the sequence $\newline$ $1,3,9,27,\dots$ , where the first term should be $\newline$ $f(1)$ . $\newline$ Khaled said the formula is $\newline$ $f(n)=1\times3^{(n-1)}$ , and $\newline$ Wilma said the formula is $\newline$ $f(n)=1\times3^{(n)}$ . $\newline$ Which one of them is right? $\newline$ Choose $1$ answer: $\newline$ (A) Only Khaled $\newline$ (B) Only Wilma $\newline$ (C) Both Khaled and Wilma $\newline$ (D) Neither Khaled nor Wilma

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Algebra 1

Write variable expressions for geometric sequences

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Q. Khaled and Wilma were asked to find an explicit formula for the sequence

\newline

1,3,9,27,\dots

, where the first term should be

\newline

f(1)

\newline

Khaled said the formula is

\newline

f(n)=1\times3^{(n-1)}

, and

\newline

Wilma said the formula is

\newline

f(n)=1\times3^{(n)}

\newline

Which one of them is right?

\newline

Choose

1

answer:

\newline

(A) Only Khaled

\newline

(B) Only Wilma

\newline

\newline

(D) Neither Khaled nor Wilma

Determine Khaled's formula: We need to determine which formula correctly represents the given geometric sequence. The sequence is $1, 3, 9, 27, \ldots$ , which means each term is multiplied by a common ratio to get the next term. Let's check Khaled's formula first.
Test Khaled's formula with first term: Khaled's formula is $f(n) = 1 \times 3^{(n-1)}$ . Let's test this formula with the first term where $n = 1$ . If Khaled is correct, $f(1)$ should equal $1$ . $\newline$ Calculation: $f(1) = 1 \times 3^{(1-1)} = 1 \times 3^0 = 1 \times 1 = 1$ . $\newline$ Since the first term of the sequence is indeed $1$ , Khaled's formula correctly calculates the first term.
Test Khaled's formula with second term: Now let's test Khaled's formula with the second term where $n = 2$ . If Khaled is correct, $f(2)$ should equal $3$ . $\newline$ Calculation: $f(2) = 1 \times 3^{(2-1)} = 1 \times 3^1 = 1 \times 3 = 3$ . $\newline$ Since the second term of the sequence is indeed $3$ , Khaled's formula correctly calculates the second term.
Test Khaled's formula with third term: Let's test Khaled's formula with the third term where $n = 3$ . If Khaled is correct, $f(3)$ should equal $9$ . $\newline$ Calculation: $f(3) = 1 \times 3^{(3-1)} = 1 \times 3^2 = 1 \times 9 = 9$ . $\newline$ Since the third term of the sequence is indeed $9$ , Khaled's formula correctly calculates the third term.
Check Wilma's formula: Now let's check Wilma's formula, which is $f(n) = 1 \times 3^n$ . We'll test this formula with the first term where $n = 1$ . If Wilma is correct, $f(1)$ should equal $1$ . $\newline$ Calculation: $f(1) = 1 \times 3^1 = 1 \times 3 = 3$ . $\newline$ Since the first term of the sequence is $1$ , not $3$ , Wilma's formula does not correctly calculate the first term.
Conclusion: Only Khaled's formula is correct: Since Wilma's formula did not correctly calculate the first term, we do not need to test it further. We can conclude that only Khaled's formula is correct.