QuestionFind a formula for $a_{n}$ based on the sequence $0, 3, -6, 9, -12, \ldots$ . What is $a_{n}=$ ?

Studdy Solution

STEP 1

Assumptions1. The sequence starts with $a0$ and continues with $a1$ , $a$ , $a3$ , $a4$ , ... . The sequence given is $0,3,-6,9,-12, \ldots$

STEP 2

First, we need to identify the pattern in the sequence.

STEP 3

By observing the sequence, we can see that the absolute values of the terms are increasing by multiples of3, and the sign alternates between positive and negative.

STEP 4

Let's write down the pattern we observe. $a0 =0 =0 \times3$ $a1 =3 =1 \times3$ $a2 = -6 =2 \times -3$ $a3 =9 =3 \times3$ $a4 = -12 =4 \times -3$

STEP 5

From the pattern, we can see that the $n$ th term is $n$ times3 or $-3$ depending on whether $n$ is even or odd.

STEP 6

We can express this pattern with the following formula $a_n = (-1)^n \times n \times3$ This formula multiplies $n$ by3 and then applies a sign change depending on whether $n$ is even (resulting in a positive value) or odd (resulting in a negative value).

STEP 7

So, the explicit formula for the $n$ th term of the sequence is $a_n = (-1)^n \times n \times3$

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QuestionFind a formula for $a_{n}$ based on the sequence $0, 3, -6, 9, -12, \ldots$ . What is $a_{n}=$ ?

Studdy Solution

STEP 1

Assumptions1. The sequence starts with $a0$ and continues with $a1$ , $a$ , $a3$ , $a4$ , ... . The sequence given is $0,3,-6,9,-12, \ldots$

STEP 2

First, we need to identify the pattern in the sequence.

STEP 3

By observing the sequence, we can see that the absolute values of the terms are increasing by multiples of3, and the sign alternates between positive and negative.

STEP 4

Let's write down the pattern we observe. $a0 =0 =0 \times3$ $a1 =3 =1 \times3$ $a2 = -6 =2 \times -3$ $a3 =9 =3 \times3$ $a4 = -12 =4 \times -3$

STEP 5

From the pattern, we can see that the $n$ th term is $n$ times3 or $-3$ depending on whether $n$ is even or odd.

STEP 6

We can express this pattern with the following formula $a_n = (-1)^n \times n \times3$ This formula multiplies $n$ by3 and then applies a sign change depending on whether $n$ is even (resulting in a positive value) or odd (resulting in a negative value).

STEP 7

So, the explicit formula for the $n$ th term of the sequence is $a_n = (-1)^n \times n \times3$

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QuestionFind a formula for ana_{n}an​ based on the sequence 0,3,−6,9,−12,…0, 3, -6, 9, -12, \ldots0,3,−6,9,−12,…. What is an=a_{n}=an​=?

Studdy Solution

STEP 1

Assumptions1. The sequence starts with a0a0a0 and continues with a1a1a1, aaa, a3a3a3, a4a4a4, ... . The sequence given is 0,3,−6,9,−12,…0,3,-6,9,-12, \ldots0,3,−6,9,−12,…

STEP 2

First, we need to identify the pattern in the sequence.

STEP 3

By observing the sequence, we can see that the absolute values of the terms are increasing by multiples of3, and the sign alternates between positive and negative.

STEP 4

Let's write down the pattern we observe.a0=0=0×3a0 =0 =0 \times3a0=0=0×3a1=3=1×3a1 =3 =1 \times3a1=3=1×3a2=−6=2×−3a2 = -6 =2 \times -3a2=−6=2×−3a3=9=3×3a3 =9 =3 \times3a3=9=3×3a4=−12=4×−3a4 = -12 =4 \times -3a4=−12=4×−3

STEP 5

From the pattern, we can see that the nnnth term is nnn times3 or −3-3−3 depending on whether nnn is even or odd.

STEP 6

We can express this pattern with the following formulaan=(−1)n×n×3a_n = (-1)^n \times n \times3an​=(−1)n×n×3This formula multiplies nnn by3 and then applies a sign change depending on whether nnn is even (resulting in a positive value) or odd (resulting in a negative value).

STEP 7

So, the explicit formula for the nnnth term of the sequence isan=(−1)n×n×3a_n = (-1)^n \times n \times3an​=(−1)n×n×3

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QuestionFind a formula for $a_{n}$ based on the sequence $0, 3, -6, 9, -12, \ldots$ . What is $a_{n}=$ ?

Assumptions1. The sequence starts with $a0$ and continues with $a1$ , $a$ , $a3$ , $a4$ , ... . The sequence given is $0,3,-6,9,-12, \ldots$

Let's write down the pattern we observe. $a0 =0 =0 \times3$ $a1 =3 =1 \times3$ $a2 = -6 =2 \times -3$ $a3 =9 =3 \times3$ $a4 = -12 =4 \times -3$

From the pattern, we can see that the $n$ th term is $n$ times3 or $-3$ depending on whether $n$ is even or odd.

We can express this pattern with the following formula $a_n = (-1)^n \times n \times3$ This formula multiplies $n$ by3 and then applies a sign change depending on whether $n$ is even (resulting in a positive value) or odd (resulting in a negative value).

So, the explicit formula for the $n$ th term of the sequence is $a_n = (-1)^n \times n \times3$