The image you provided appears to be a worksheet or assessment related to
"Arithmetic Sequences and Series". The task involves solving problems related to arithmetic sequences, specifically finding the sum of terms in an arithmetic sequence.
Problem Statement:
The problem asks us to find the sum of the first 10 terms of an arithmetic sequence where:
- The first term (\(a_1\)) is 3.
- The common difference (\(d\)) is 4.
Solution:
#### Step 1: Recall the formula for the sum of the first \(n\) terms of an arithmetic sequence
The sum of the first \(n\) terms of an arithmetic sequence is given by:
\[
S_n = \frac{n}{2} \left( 2a_1 + (n-1)d \right)
\]
where:
- \(S_n\) is the sum of the first \(n\) terms,
- \(a_1\) is the first term,
- \(d\) is the common difference,
- \(n\) is the number of terms.
#### Step 2: Identify the given values
From the problem:
- \(a_1 = 3\),
- \(d = 4\),
- \(n = 10\).
#### Step 3: Substitute the values into the formula
Substitute \(a_1 = 3\), \(d = 4\), and \(n = 10\) into the formula:
\[
S_{10} = \frac{10}{2} \left( 2 \cdot 3 + (10-1) \cdot 4 \right)
\]
#### Step 4: Simplify the expression inside the parentheses
First, calculate \(2 \cdot 3\):
\[
2 \cdot 3 = 6
\]
Next, calculate \((10-1) \cdot 4\):
\[
(10-1) \cdot 4 = 9 \cdot 4 = 36
\]
Add these results together:
\[
6 + 36 = 42
\]
#### Step 5: Complete the calculation
Now, substitute back into the formula:
\[
S_{10} = \frac{10}{2} \cdot 42
\]
Simplify \(\frac{10}{2}\):
\[
\frac{10}{2} = 5
\]
Multiply:
\[
S_{10} = 5 \cdot 42 = 210
\]
Final Answer:
The sum of the first 10 terms of the arithmetic sequence is:
\[
\boxed{210}
\]
Explanation:
- We used the formula for the sum of an arithmetic sequence, which accounts for the first term, the common difference, and the number of terms.
- By substituting the given values and simplifying step-by-step, we arrived at the final result.
If you have any further questions or need additional clarification, feel free to ask!
Parent Tip: Review the logic above to help your child master the concept of transport in plants worksheet answers.
