Find the Term-to-Term Rule: Sequence 51,47,43,39

Q: What's the difference between term-to-term and position-to-term rules?

Term-to-term rule tells you how to get from one term to the next (subtract 4). Position-to-term rule is the formula $ a_n = 55 - 4n $ that lets you find any term directly!

Q: Why do I get 55 in the formula when the first term is 51?

When you expand $ a_n = 51 + (n-1)(-4) $, you get $ a_n = 51 - 4n + 4 = 55 - 4n $. The 55 comes from combining 51 + 4, not from the original sequence.

Q: How do I know if the common difference is positive or negative?

Look at the sequence direction! If terms get smaller (like 51, 47, 43...), the common difference is negative. If they get larger, it's positive.

Q: Can I check my formula works for all the given terms?

Yes! Test each term: $ a_1 = 55-4(1) = 51 $ ✓, $ a_2 = 55-4(2) = 47 $ ✓, $ a_3 = 55-4(3) = 43 $ ✓. All match!

Q: What if I need to find the 10th term?

Just substitute n = 10 into your formula: $ a_{10} = 55 - 4(10) = 55 - 40 = 15 $. That's the power of having the position-to-term rule!

Arithmetic Sequences with Negative Common Difference

What is the term-to-term rule for the sequence below?

$51,47,43,39\ldots$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the sequence formula

00:03 Let's look at the constant difference between each term

00:13 Let's pay attention to the first term

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

What is the term-to-term rule for the sequence below?

$51,47,43,39\ldots$

Step-by-step solution

To determine the term-to-term rule for this sequence, we need to identify the pattern of change between terms. In this sequence, each term is obtained by subtracting 4 from the previous term:

The first term is 51.
The second term is 47, which is 51 - 4.
The third term is 43, which is 47 - 4.
The fourth term is 39, which is 43 - 4.

Thus, the common difference, $d$ , between consecutive terms is $-4$ .

We can use the formula for the $n$ -th term of an arithmetic sequence:

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

Here, $a_1 = 51$ and $d = -4$ . Substituting these into the formula gives:

$a_n = 51 + (n-1) \times (-4)$

Expanding this equation, we have:

$a_n = 51 - 4(n-1)$

Simplifying, we get:

$a_n = 51 - 4n + 4$

$a_n = 55 - 4n$

Therefore, the term-to-term rule for this sequence is $a_n = 55 - 4n$ .

Final Answer

$an=55-4n$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Each term decreases by the same amount (common difference)
Formula Application: Use $a_n = a_1 + (n-1)d$ where d = -4
Verification: Check by substituting: $a_2 = 55 - 4(2) = 47$ ✓

Common Mistakes

Avoid these frequent errors

Confusing term-to-term with position-to-term rule
Don't just say 'subtract 4' as the final answer = incomplete solution! The question asks for the algebraic formula, not just the pattern. Always find the position-to-term rule $a_n = 55 - 4n$ that gives any term's value.

Practice Quiz

Test your knowledge with interactive questions

Is there a term-to-term rule for the sequence below?

18 , 22 , 26 , 30

FAQ

Everything you need to know about this question

What's the difference between term-to-term and position-to-term rules?

Term-to-term rule tells you how to get from one term to the next (subtract 4). Position-to-term rule is the formula $a_n = 55 - 4n$ that lets you find any term directly!

Why do I get 55 in the formula when the first term is 51?

When you expand $a_n = 51 + (n-1)(-4)$ , you get $a_n = 51 - 4n + 4 = 55 - 4n$ . The 55 comes from combining 51 + 4, not from the original sequence.

How do I know if the common difference is positive or negative?

Look at the sequence direction! If terms get smaller (like 51, 47, 43...), the common difference is negative. If they get larger, it's positive.

Can I check my formula works for all the given terms?

Yes! Test each term: $a_1 = 55-4(1) = 51$ ✓, $a_2 = 55-4(2) = 47$ ✓, $a_3 = 55-4(3) = 43$ ✓. All match!

What if I need to find the 10th term?

Just substitute n = 10 into your formula: $a_{10} = 55 - 4(10) = 55 - 40 = 15$ . That's the power of having the position-to-term rule!

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