QuestionFind a formula for the $n^{\text{th}}$ term $a_{n}$ of the sequence $3, -3, -9, \ldots$

Studdy Solution

STEP 1

Assumptions1. The sequence given is $3, -3, -9, \ldots$ . We need to find an explicit formula for $a_{n}$ , the $n^{\text {th }}$ term of the sequence.

STEP 2

First, we need to identify the pattern in the sequence.The sequence starts at and then alternates sign and multiplies by.

STEP 3

The sequence can be written as $3, -3 \cdot1, -3 \cdot3, \ldots$

STEP 4

We can see that the absolute value of each term is a power of3.The first term is $3^1$ , the second term is $3^1$ , the third term is $3^2$ , and so on.

STEP 5

We can also see that the sign of each term alternates.The first term is positive, the second term is negative, the third term is negative, the fourth term is positive, and so on.

STEP 6

We can combine these observations to write a general formula for the $n^{\text {th }}$ term of the sequence.

STEP 7

The general formula for the $n^{\text {th }}$ term of the sequence is $a_{n} = (-1)^{n+1} \cdot3^n$ This formula captures the alternating sign and the powers of3.

STEP 8

Let's check this formula with the given sequence.
For $n=1$ , $a_{1} = (-1)^{1+1} \cdot3^1 =3$ .
For $n=2$ , $a_{2} = (-1)^{2+1} \cdot3^2 = -3$ .
For $n=3$ , $a_{3} = (-1)^{3+1} \cdot3^3 = -$ .
The formula matches the given sequence, so it is correct.
The explicit formula for $a_{n}$ , the $n^{\text {th }}$ term of the sequence $3, -3, -, \ldots$ is $a_{n} = (-1)^{n+1} \cdot3^n$ .

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