QuestionFind the 76 th term of the arithmetic sequence $16,14,12, \ldots$

Studdy Solution

STEP 1

What is this asking? We need to find the 76th number in a sequence that starts at 16 and decreases by 2 each time. Watch out! It's easy to mess up the signs when the sequence is decreasing, so let's be super careful!

STEP 2

1. Find the common difference.
2. Find the formula for the nth term.
3. Calculate the 76th term.

STEP 3

Let's look at the difference between consecutive terms in our sequence.
The difference between the second term, $14$ , and the first term, $16$ , is $14 - 16 = -2$ .

STEP 4

The difference between the third term, $12$ , and the second term, $14$ , is $12 - 14 = -2$ .
So, our common difference, often denoted as $d$ , is $-2$ .
This tells us how much the sequence changes from one term to the next.

STEP 5

The general formula for the nth term of an arithmetic sequence is given by $a_n = a_1 + (n - 1) \cdot d$ , where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number we're looking for, and $d$ is the common difference.

STEP 6

In our case, the first term $a_1$ is $16$ , and we already found that our common difference $d$ is $-2$ .
So, our formula becomes $a_n = 16 + (n - 1) \cdot (-2)$ .

STEP 7

We want to find the 76th term, so we'll plug in $n = 76$ into our formula: $a_{76} = 16 + (76 - 1) \cdot (-2)$ .

STEP 8

Let's simplify inside the parentheses first: $76 - 1 = 75$ .
So, our expression becomes $a_{76} = 16 + 75 \cdot (-2)$ .

STEP 9

Now, we multiply: $75 \cdot (-2) = -150$ .
Our expression is now $a_{76} = 16 + (-150)$ .

STEP 10

Finally, we add: $16 + (-150) = -134$ .
So, the 76th term of our sequence is $-134$ !

STEP 11

The 76th term of the arithmetic sequence is $-134$ .

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