Number Sequence Analysis: Is -1 in the Series 51, 47, 43, 39, ...?

Q: Can arithmetic sequences have negative numbers?

Absolutely! Arithmetic sequences can include negative numbers, zero, and positive numbers. The pattern continues forever in both directions as long as you keep adding (or subtracting) the common difference.

Q: What if I get a decimal when solving for n?

If n is not a whole number, then that value is not part of the sequence. Sequence positions must be positive integers: n = 1, 2, 3, 4, etc.

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

Look at how the sequence changes: if terms get smaller (like 51 → 47 → 43), the common difference is negative. If they get larger, it's positive.

Q: Why do we use the formula instead of just listing terms?

Listing terms takes forever for large sequences! The formula $ a_n = a_1 + (n-1)d $ lets you jump directly to any term without writing out all the ones before it.

Arithmetic Sequences with Negative Terms

Assuming that the series continues with the same legality, does the number $-1$ Is it part of the series?

$51,47,43,39\ldots$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Is (-1) a member of the sequence?

00:03 This is the sequence formula

00:07 Let's substitute in the formula and solve for X

00:11 If the solution for X is positive and whole, then it's a member of the sequence

00:16 Let's isolate X

00:44 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

Assuming that the series continues with the same legality, does the number $-1$ Is it part of the series?

$51,47,43,39\ldots$

Step-by-step solution

We are asked to determine whether the number $-1$ is part of the sequence $51, 47, 43, 39, \ldots$ .

This is an arithmetic sequence where the first term $a_1 = 51$ and the common difference $d = -4$ .

The formula for the $n$ -th term of an arithmetic sequence is given by: $a_n = a_1 + (n-1) \cdot d$

Substituting the known values, we get: $a_n = 51 + (n-1)(-4)$

We want to find $n$ such that $a_n = -1$ . Thus, $-1 = 51 + (n-1)(-4)$

Expanding and simplifying yields: $-1 = 51 - 4n + 4$
$-1 = 55 - 4n$
$4n = 55 + 1$
$4n = 56$
$n = \frac{56}{4}$
$n = 14$

Since $n = 14$ is a positive integer, $-1$ is indeed part of the sequence, appearing as the 14th term.

Therefore, the correct answer to the problem is Yes.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Identify first term and common difference from sequence
Formula Application: Use $a_n = 51 + (n-1)(-4)$ to find position
Verification: Check that n = 14 gives integer position ✓

Common Mistakes

Avoid these frequent errors

Assuming negative numbers can't be in arithmetic sequences
Don't think sequences must stay positive = missing valid terms! Arithmetic sequences continue infinitely in both directions with the same pattern. Always solve the equation even if the target is negative.

Practice Quiz

Test your knowledge with interactive questions

12 ☐ 10 ☐ 8 7 6 5 4 3 2 1

Which numbers are missing from the sequence so that the sequence has a term-to-term rule?

FAQ

Everything you need to know about this question

Can arithmetic sequences have negative numbers?

Absolutely! Arithmetic sequences can include negative numbers, zero, and positive numbers. The pattern continues forever in both directions as long as you keep adding (or subtracting) the common difference.

What if I get a decimal when solving for n?

If n is not a whole number, then that value is not part of the sequence. Sequence positions must be positive integers: n = 1, 2, 3, 4, etc.

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

Look at how the sequence changes: if terms get smaller (like 51 → 47 → 43), the common difference is negative. If they get larger, it's positive.

Why do we use the formula instead of just listing terms?

Listing terms takes forever for large sequences! The formula $a_n = a_1 + (n-1)d$ lets you jump directly to any term without writing out all the ones before it.

Can I check my answer by counting backwards from 51?

Yes! You can verify: 51, 47, 43, 39, 35, 31, 27, 23, 19, 15, 11, 7, 3, -1. Count the positions to confirm -1 is the 14th term.

What happens if the sequence never reaches the target number?

Sometimes the target number doesn't fit the pattern. If solving for n gives a decimal or negative number, then that value isn't part of the sequence.

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