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

Q: How do I know if the sequence is increasing or decreasing?

Look at the common difference! If d is positive, the sequence increases. If d is negative (like -3 in this problem), the sequence decreases. The series 16, 13, 10, 7 goes down by 3 each time.

Q: Why do we use (n-1) instead of just n in the formula?

The formula $ a_n = a_1 + (n-1)d $ accounts for the fact that we start counting positions from n=1. When n=1, we get $ a_1 + (1-1)d = a_1 $, which gives us the first term correctly.

Q: Can I check my formula with any term in the sequence?

Yes! Pick any position and substitute. For example, the 4th term should be 7. Using $ 19-3n $: when n=4, we get 19-3(4) = 19-12 = 7 ✓

Q: What if I get a different formula that still works?

Sometimes equivalent forms exist, but for arithmetic sequences, there's usually one standard form. Make sure your formula gives the correct values for all given terms before considering it correct.

Q: How do I simplify the final expression correctly?

After substituting into $ a_n = 16 + (n-1)(-3) $, distribute carefully: $ 16 - 3n + 3 $. Then combine like terms: $ 16 + 3 - 3n = 19 - 3n $.

Arithmetic Sequences with Formula Derivation

Write as a formula the legality of the series:

16,13,10,7

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

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 written solution

Follow each step carefully to understand the complete solution

Understand the problem

Write as a formula the legality of the series:

16,13,10,7

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 $19 - 3n$ .

Final Answer

$19-3n$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Find common difference by subtracting consecutive terms
Formula Application: Use $a_n = a_1 + (n-1)d$ where d = -3
Verification: Check n=1: 19-3(1)=16, n=2: 19-3(2)=13 ✓

Common Mistakes

Avoid these frequent errors

Using addition instead of subtraction for common difference
Don't calculate d = 16 - 13 = 3 instead of d = 13 - 16 = -3! This gives the wrong sign and creates an increasing sequence instead of decreasing. Always subtract first term from second term to get the correct common difference.

Practice Quiz

Test your knowledge with interactive questions

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 94,96,98,100,102,104 $$

FAQ

Everything you need to know about this question

How do I know if the sequence is increasing or decreasing?

Look at the common difference! If d is positive, the sequence increases. If d is negative (like -3 in this problem), the sequence decreases. The series 16, 13, 10, 7 goes down by 3 each time.

Why do we use (n-1) instead of just n in the formula?

The formula $a_n = a_1 + (n-1)d$ accounts for the fact that we start counting positions from n=1. When n=1, we get $a_1 + (1-1)d = a_1$ , which gives us the first term correctly.

Can I check my formula with any term in the sequence?

Yes! Pick any position and substitute. For example, the 4th term should be 7. Using $19-3n$ : when n=4, we get 19-3(4) = 19-12 = 7 ✓

What if I get a different formula that still works?

Sometimes equivalent forms exist, but for arithmetic sequences, there's usually one standard form. Make sure your formula gives the correct values for all given terms before considering it correct.

How do I simplify the final expression correctly?

After substituting into $a_n = 16 + (n-1)(-3)$ , distribute carefully: $16 - 3n + 3$ . Then combine like terms: $16 + 3 - 3n = 19 - 3n$ .

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