Derive the Series Formula: Decoding 16, 13, 10, 7

Video Solution

Solution Steps

00:00 Find the sequence formula

00:04 Identify the first term according to the given data

00:15 Note the constant difference between terms

00:19 This is the constant difference

00:26 Use the formula to describe an arithmetic sequence

00:32 Substitute appropriate values and solve to find the sequence formula

00:46 Properly open parentheses, multiply by each factor

00:53 Continue solving

01:03 And this is the solution to the question

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Identify the first term and the common difference of the arithmetic sequence.
Step 2: Apply the nth term formula for arithmetic sequences.
Step 3: Simplify the expression to match the provided answer choices.

Now, let's work through each step:

Step 1: Identify the first term and common difference:

The first term $a_1$ of the sequence is 16. To find the common difference $d$ , subtract the second term from the first term:

$d = 13 - 16 = -3$

Step 2: Apply the nth term formula for arithmetic sequences:

The formula for the nth term of an arithmetic sequence is:

$a_n = a_1 + (n-1) \times d$

Substitute $a_1 = 16$ and $d = -3$ :

$a_n = 16 + (n-1)(-3)$

Step 3: Simplify the expression:

Distribute $(-3)$ across the $(n-1)$ :

$a_n = 16 - 3n + 3$

Simplify further:

$a_n = 19 - 3n$

Thus, the correct formula for the series is:

$19 - 3n$

Therefore, the solution to the problem is $19 - 3n$ .