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

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

18 , 22 , 26 , 30

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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