Question8. Write the explicit and recursive formula for the sequence $-4,-1,2, \ldots$

Studdy Solution

STEP 1

What is this asking? We need to find two different ways to describe this sequence of numbers: one that lets us jump straight to any number in the sequence (explicit), and one that builds each number from the one before it (recursive). Watch out! Don't mix up the explicit and recursive formulas!
They have different structures and purposes.
Also, be careful with the signs – it's easy to make a mistake with those pesky negatives!

STEP 2

1. Find the common difference
2. Write the recursive formula
3. Write the explicit formula

STEP 3

Let's see how much the sequence changes between each term.
The difference between the second term and the first term is $(-1) - (-4)$ .
Remember subtracting a negative is the same as adding a positive, so $(-1) - (-4) = -1 + 4 = 3$ .
The difference between the third term and the second term is $2 - (-1) = 2 + 1 = 3$ .

STEP 4

Since the difference between consecutive terms is consistently 3, this is our common difference, often denoted as $d$ .
So, $d = 3$ .
This tells us that each term in the sequence is 3 greater than the previous term.

STEP 5

The recursive formula needs a starting point.
Our initial term, often called $a_1$ , is the first term in the sequence, which is $-4$ .
So, $a_1 = -4$ .

STEP 6

The recursive formula tells us how to get any term in the sequence based on the term before it.
Since each term is 3 greater than the previous term, we can write the recursive formula as $a_n = a_{n-1} + 3$ , where $a_n$ is the $n$ -th term and $a_{n-1}$ is the term right before it.

STEP 7

Putting it all together, the recursive formula is $a_1 = -4$ and $a_n = a_{n-1} + 3$ for $n > 1$ .

STEP 8

The explicit formula for an arithmetic sequence is given by $a_n = a_1 + (n-1) \cdot d$ , where $a_n$ is the $n$ -th term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

STEP 9

We already know $a_1 = -4$ and $d = 3$ .
Substituting these values into the general formula, we get $a_n = -4 + (n-1) \cdot 3$ .

STEP 10

Let's distribute the 3: $a_n = -4 + 3n - 3$ .
Combining the constant terms, we get $a_n = 3n - 7$ .

STEP 11

The recursive formula for the sequence is $a_1 = -4$ and $a_n = a_{n-1} + 3$ for $n > 1$ .
The explicit formula is $a_n = 3n - 7$ .

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