Question Find the common difference and recursive/explicit formulas for the sequence $12, 16, 20, 24, 28, \ldots$ .

Studdy Solution

STEP 1

Assumptions
1. The given sequence is arithmetic, which means it has a constant difference between consecutive terms.
2. We are asked to find the common difference, the first term of the sequence $f(1)$ , the recursive formula, and the explicit formula for the sequence.

STEP 2

To find the common difference, we subtract any term in the sequence from the subsequent term.
$Common\, difference = Term_{n+1} - Term_{n}$

STEP 3

Using the first two terms of the sequence, calculate the common difference.
$Common\, difference = 16 - 12$

STEP 4

Calculate the common difference.
$Common\, difference = 4$

STEP 5

The first term of the sequence, denoted by $f(1)$ , is simply the first number in the sequence.
$f(1) = 12$

STEP 6

The recursive formula for an arithmetic sequence is given by:
$f(n) = f(n-1) + d$
where $d$ is the common difference.

STEP 7

We know the common difference $d$ is 4, so we can write the recursive formula as:
$f(n) = f(n-1) + 4$

STEP 8

The explicit formula for an arithmetic sequence is given by:
$f(n) = a + (n-1)d$
where $a$ is the first term and $d$ is the common difference.

STEP 9

We know the first term $a$ is 12 and the common difference $d$ is 4, so we can write the explicit formula as:
$f(n) = 12 + (n-1) \cdot 4$

STEP 10

Simplify the explicit formula.
$f(n) = 12 + 4n - 4$

STEP 11

Combine like terms in the explicit formula.
$f(n) = 4n + 8$
The common difference is 4, the first term $f(1)$ is 12, the recursive formula is $f(n) = f(n-1) + 4$ , and the explicit formula is $f(n) = 4n + 8$ .

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