Find the Variable Expression for Sequence: 3, 8.5, 14, 19.5, 25

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

Because when n = 1, we want the first term exactly, with zero jumps from the starting position. Using (n-1) gives us (1-1) = 0 jumps, so a(1) = 3 + 0 × 5.5 = 3 ✓

Q: How do I find the common difference when I have decimals?

Subtract any term from the next term: 8.5 - 3 = 5.5, or 14 - 8.5 = 5.5. The difference should be the same between all consecutive terms.

Q: What if I want to find the 10th term in this sequence?

Use the formula: $ a(10) = 3 + (10-1) \times 5.5 = 3 + 9 \times 5.5 = 3 + 49.5 = 52.5 $

Q: Can arithmetic sequences have negative common differences?

Yes! If each term gets smaller, the common difference is negative. For example: 20, 15, 10, 5 has d = -5.

Q: How do I check if my formula is correct?

Test it with at least 2-3 known terms from the sequence. If your formula gives the right values for those positions, it's correct!

Q: What does the 'n' represent in the formula?

The variable n represents the position of the term in the sequence. So n=1 is the 1st term, n=2 is the 2nd term, and so on.

Arithmetic Sequences with Decimal Differences

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

$3,8.5,14,19.5,25$

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

00:08 Let's observe the change between terms (D) according to the given data

00:29 We'll use the formula to describe the sequence

00:37 We'll substitute appropriate values and solve to find the sequence formula

00:51 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:

$3,8.5,14,19.5,25$

Describe the property using the variable $n$

Step-by-step solution

To solve this problem, let's derive the formula for the arithmetic sequence:

Step 1: Identify the first term $a_1$ . In this series, the first term is $3$ .
Step 2: Determine the common difference $d$ . By calculation, the difference between each consecutive term is constant at $5.5$ .
Step 3: Use the common formula for an arithmetic sequence: $a(n) = a_1 + (n-1) \cdot d$ .
Step 4: Substitute the values into the formula: $a(n) = 3 + (n-1) \cdot 5.5$ .

Applying these steps confirms the general term for this arithmetic sequence is:

$a(n) = 3 + (n-1) \times 5.5$

Thus, the correct formulation for $n$ -th term of the sequence is given by this expression.

Therefore, the correct answer is choice 3: $a(n)=3+(n-1)\times5.5$ .

Final Answer

$a(n)=3+(n-1)\times5.5$

Key Points to Remember

Essential concepts to master this topic

Formula: For arithmetic sequences use $a(n) = a_1 + (n-1) \cdot d$
Method: Find first term a₁ = 3 and common difference d = 5.5
Verify: Check formula with known terms: a(2) = 3 + (2-1) × 5.5 = 8.5 ✓

Common Mistakes

Avoid these frequent errors

Using n instead of (n-1) in the formula
Don't write a(n) = 3 + n × 5.5 = gives wrong values for all positions! This formula makes the first term equal 8.5 instead of 3. Always use (n-1) because when n=1, we want zero additional jumps from the first term.

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

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

Because when n = 1, we want the first term exactly, with zero jumps from the starting position. Using (n-1) gives us (1-1) = 0 jumps, so a(1) = 3 + 0 × 5.5 = 3 ✓

How do I find the common difference when I have decimals?

Subtract any term from the next term: 8.5 - 3 = 5.5, or 14 - 8.5 = 5.5. The difference should be the same between all consecutive terms.

What if I want to find the 10th term in this sequence?

Use the formula: $a(10) = 3 + (10-1) \times 5.5 = 3 + 9 \times 5.5 = 3 + 49.5 = 52.5$

Can arithmetic sequences have negative common differences?

Yes! If each term gets smaller, the common difference is negative. For example: 20, 15, 10, 5 has d = -5.

How do I check if my formula is correct?

Test it with at least 2-3 known terms from the sequence. If your formula gives the right values for those positions, it's correct!

What does the 'n' represent in the formula?

The variable n represents the position of the term in the sequence. So n=1 is the 1st term, n=2 is the 2nd term, and so on.

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