Manoj and Shira were asked to find an explicit formula for the sequence $\newline$ $-9,-27,-81,-243,\dots$ , where the first term should be $\newline$ $h(1)$ . $\newline$ Manoj said the formula is $\newline$ $h(n)=-9\cdot3^{(n-1)}$ , and $\newline$ Shira said the formula is $\newline$ $h(n)=-3\cdot3^{(n-1)}$ . $\newline$ Which one of them is right? $\newline$ Choose $1$ answer: $\newline$ (A) Only Manoj $\newline$ (B) Only Shira $\newline$ (C) Both Manoj and Shira $\newline$ (D) Neither Manoj nor Shira

View step-by-step help

Home

Math Problems

Algebra 1

Write variable expressions for geometric sequences

Full solution

Q. Manoj and Shira were asked to find an explicit formula for the sequence

\newline

-9,-27,-81,-243,\dots

, where the first term should be

\newline

h(1)

\newline

Manoj said the formula is

\newline

h(n)=-9\cdot3^{(n-1)}

, and

\newline

Shira said the formula is

\newline

h(n)=-3\cdot3^{(n-1)}

\newline

Which one of them is right?

\newline

Choose

1

answer:

\newline

(A) Only Manoj

\newline

(B) Only Shira

\newline

\newline

(D) Neither Manoj nor Shira

Identifying Geometric Sequence: The sequence given is $-9, -27, -81, -243, \ldots$ which is a geometric sequence because each term is obtained by multiplying the previous term by a common ratio.
Finding the Common Ratio: To find the common ratio $r$ , we divide the second term by the first term: $r = \frac{-27}{-9} = 3$ .
Writing the Explicit Formula: The first term of the sequence is $h(1) = -9$ . Using the common ratio $r = 3$ , we can write the explicit formula for the $n$ th term as $h(n) = h(1) \cdot r^{(n-1)}$ .
Checking Manoj's Formula: Substituting the values of $h(1)$ and $r$ into the formula, we get $h(n) = -9 \times 3^{(n-1)}$ .
Checking Shira's Formula: Now let's check Manoj's formula: $h(n) = -9 \times 3^n$ . This formula would give us the sequence $-9$ , $-27$ , $-81$ , $-243$ , ... if we start counting from $n = 0$ . However, we are asked to start from $n = 1$ , so Manoj's formula is incorrect for this sequence.
Determining the Correct Answer: Let's check Shira's formula: $h(n) = -3 \cdot 3^n$ . If we substitute $n = 1$ , we get $h(1) = -3 \cdot 3^1 = -9$ , which is the correct first term. However, for $n = 2$ , we get $h(2) = -3 \cdot 3^2 = -27$ , which is also the correct second term. But the formula should be $h(n) = -3 \cdot 3^{(n-1)}$ to match the sequence starting from $n = 1$ . Therefore, Shira's formula is also incorrect.
Determining the Correct Answer: Let's check Shira's formula: $h(n) = -3 \cdot 3^n$ . If we substitute $n = 1$ , we get $h(1) = -3 \cdot 3^1 = -9$ , which is the correct first term. However, for $n = 2$ , we get $h(2) = -3 \cdot 3^2 = -27$ , which is also the correct second term. But the formula should be $h(n) = -3 \cdot 3^{(n-1)}$ to match the sequence starting from $n = 1$ . Therefore, Shira's formula is also incorrect.Since neither Manoj's nor Shira's formula correctly represents the sequence when starting from $n = 1$ , the correct answer is (D) Neither Manoj nor Shira.