Find the nth Term Formula for Sequence: 7, 11, 15, 19, 23

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

Because the first term is at position n=1, not n=0! When n=1, we want $ a(1) = a_1 + 0 \times d = a_1 $. Using (n-1) gives us 0 when n=1, which is exactly what we need.

Q: How do I find the common difference?

Subtract any term from the next term: 11 - 7 = 4, or 15 - 11 = 4, or 19 - 15 = 4. The difference should be the same between all consecutive pairs!

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

Just substitute n=10 into your formula: $ a(10) = 7 + (10-1) \times 4 = 7 + 36 = 43 $. No need to list all terms!

Q: Can I simplify the formula a(n) = 7 + (n-1) × 4?

Yes! Expand it: $ a(n) = 7 + 4n - 4 = 4n + 3 $. Both forms are correct, but 4n + 3 is often easier to use for calculations.

Q: How do I know if a sequence is arithmetic?

Check if the difference between consecutive terms is constant. In this sequence: 11-7=4, 15-11=4, 19-15=4, 23-19=4. Since all differences equal 4, it's arithmetic!

Arithmetic Sequences with Common Difference

Given the series whose difference between two jumped numbers is constant:

$7,11,15,19,23$

Describe the property using the variable $n$

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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 This is the first term according to the given data

00:11 Let's observe the change between terms (D) according to the given data

00:28 This is the constant difference in the sequence (D)

00:33 Let's use the formula to describe the sequence

00:40 Let's substitute appropriate values and solve to find the sequence formula

00:48 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

Given the series whose difference between two jumped numbers is constant:

$7,11,15,19,23$

Describe the property using the variable $n$

Step-by-step solution

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

Step 1: Identify the first term and the common difference.
Step 2: Use the arithmetic sequence formula to express the series.
Step 3: Construct the formula based on the calculations.

Now, let's work through each step:
Step 1: The first term $a_1$ of the series is $7$ . Calculate the common difference $d$ as the difference between two consecutive terms. Between $7$ and $11$ , the difference is $4$ , and this holds for each consecutive pair of terms. Thus, $d = 4$ .

Step 2: We'll use the formula for the $n$ -th term of an arithmetic sequence, which is $a(n) = a_1 + (n-1) \times d$ .

Step 3: Substitute the values for $a_1$ and $d$ into the formula:
$a(n) = 7 + (n-1) \times 4$ .

Therefore, the solution to the problem is $a(n)=7+(n-1)\times4$ .

Final Answer

$a(n)=7+(n-1)\times4$

Key Points to Remember

Essential concepts to master this topic

Pattern: Arithmetic sequences have constant difference between consecutive terms
Formula: Use $a(n) = a_1 + (n-1) \times d$ where d = 4
Verify: Check that a(3) = 7 + (3-1) × 4 = 15 ✓

Common Mistakes

Avoid these frequent errors

Using wrong position multiplier in formula
Don't use (n+1) or just n in the formula = positions shifted incorrectly! This makes your formula give wrong values for each term position. Always use (n-1) because the first term is at position 1, not 0.

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

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

Because the first term is at position n=1, not n=0! When n=1, we want $a(1) = a_1 + 0 \times d = a_1$ . Using (n-1) gives us 0 when n=1, which is exactly what we need.

How do I find the common difference?

Subtract any term from the next term: 11 - 7 = 4, or 15 - 11 = 4, or 19 - 15 = 4. The difference should be the same between all consecutive pairs!

What if I want to find the 10th term quickly?

Just substitute n=10 into your formula: $a(10) = 7 + (10-1) \times 4 = 7 + 36 = 43$ . No need to list all terms!

Can I simplify the formula a(n) = 7 + (n-1) × 4?

Yes! Expand it: $a(n) = 7 + 4n - 4 = 4n + 3$ . Both forms are correct, but 4n + 3 is often easier to use for calculations.

How do I know if a sequence is arithmetic?

Check if the difference between consecutive terms is constant. In this sequence: 11-7=4, 15-11=4, 19-15=4, 23-19=4. Since all differences equal 4, it's arithmetic!

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