Arithmetic Sequence Problem: Finding Terms Before 10 with -3.5 Difference

Q: Why is the common difference negative in a descending sequence?

In a descending sequence, each term is smaller than the previous one. Since we subtract 3.5 from each term to get the next, the common difference is $ d = -3.5 $.

Q: How do I find terms that come before a given term?

Use the formula $ a_n = a_1 + (n-1) \cdot d $ and solve for $ a_1 $. Then use $ a_1 $ to find $ a_2 $, and so on.

Q: Can I just add 3.5 to go backwards in the sequence?

Yes! Since the common difference is -3.5, going backwards means adding 3.5. So from $ a_3 = 10 $: $ a_2 = 10 + 3.5 = 13.5 $ and $ a_1 = 13.5 + 3.5 = 17 $.

Arithmetic Sequences with Negative Common Differences

Given a descending series. The third element is 10, each element of the series is smaller in 3.5 than its predecessor.

Select the first and second element of the series.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the first and second terms in the sequence

00:08 The difference between each term according to the given data

00:25 Let's calculate the number before 10

00:33 We'll use the same method to find the first term

00:47 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 a descending series. The third element is 10, each element of the series is smaller in 3.5 than its predecessor.

Select the first and second element of the series.

Step-by-step solution

To solve this problem, we will take the following steps:

Use the information $a_3 = 10$ and the common difference $d = -3.5$ to find the earlier terms in the sequence.
Apply the arithmetic sequence formula: $a_n = a_1 + (n-1) \cdot d$ .

Let's apply these steps:

Step 1: Using the given formula for the $n$ -th term in an arithmetic sequence, $a_3 = a_1 + (3-1) \cdot d = 10.$

Plug in the common difference $d = -3.5$ :

$10 = a_1 + 2 \cdot (-3.5).$

Simplify the above equation:

$10 = a_1 - 7.$

Solving for $a_1$ , we get:

$a_1 = 10 + 7 = 17.$

Step 2: To find $a_2$ , use:

$a_2 = a_1 + (2-1) \cdot (-3.5) = 17 - 3.5.$

Thus, $a_2 = 17 - 3.5 = 13.5.$

Therefore, the first element is 17, and the second element is 13.5.

The solution to the problem is $13.5, 17$ .

Final Answer

13.5 , 17

Key Points to Remember

Essential concepts to master this topic

Formula: Use $a_n = a_1 + (n-1) \cdot d$ to find any term
Technique: Work backwards: if $a_3 = 10$ and $d = -3.5$ , then $a_1 = 17$
Check: Verify sequence: 17, 13.5, 10 decreases by 3.5 each step ✓

Common Mistakes

Avoid these frequent errors

Adding the common difference instead of subtracting
Don't add 3.5 to find earlier terms = going the wrong direction! This makes the sequence increase when it should decrease. Always remember that negative common difference means subtract the absolute value when moving forward, or add when going backward.

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 is the common difference negative in a descending sequence?

In a descending sequence, each term is smaller than the previous one. Since we subtract 3.5 from each term to get the next, the common difference is $d = -3.5$ .

How do I find terms that come before a given term?

Use the formula $a_n = a_1 + (n-1) \cdot d$ and solve for $a_1$ . Then use $a_1$ to find $a_2$ , and so on.

Can I just add 3.5 to go backwards in the sequence?

Yes! Since the common difference is -3.5, going backwards means adding 3.5. So from $a_3 = 10$ : $a_2 = 10 + 3.5 = 13.5$ and $a_1 = 13.5 + 3.5 = 17$ .

How do I check if my sequence is correct?

Write out your terms and verify the pattern. Each term should be exactly 3.5 less than the previous term: 17 → 13.5 → 10. The differences should all equal -3.5.

What if the question asks for terms in a different order?

Read carefully! This question asks for "first and second element," so the answer is $a_1$ first, then $a_2$ : 17, 13.5. But the correct choice shows 13.5, 17.

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