Find the sum of the terms of arithmetic progression ; 2+4+6+......+152

To find the sum of the terms of an arithmetic progression (AP), we can use the formula:

\[
S_n = \frac{n}{2} \times (a + l)
\]

where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( n \) is the number of terms,
- \( a \) is the first term, and
- \( l \) is the last term.

### Step 1: Identify the values
In the given AP, we have:
- First term, \( a = 2 \)
- Last term, \( l = 152 \)
- Common difference, \( d = 4 - 2 = 2 \)

### Step 2: Find the number of terms, \( n \)
The formula to find the number of terms in an AP is:

\[
l = a + (n - 1) \times d
\]

Substitute the values:

\[
152 = 2 + (n - 1) \times 2
\]

Simplify and solve for \( n \):

\[
152 = 2 + 2n - 2
\]
\[
152 = 2n
\]
\[
n = \frac{152}{2} = 76
\]

### Step 3: Find the sum of the first \( n \) terms
Now that we have \( n = 76 \), we can find the sum using the sum formula:

\[
S_{76} = \frac{76}{2} \times (2 + 152)
\]
\[
S_{76} = 38 \times 154
\]
\[
S_{76} = 5852
\]

So, the sum of the terms of the arithmetic progression is \( \boxed{5852} \).

Would you like further details or have any questions about this?  

Here are some related questions:
1. How do you determine the common difference in an arithmetic progression?
2. What is the formula for finding the nth term of an arithmetic sequence?
3. How can you find the sum of the first n terms when the last term is not given?
4. What are some real-life examples where arithmetic sequences appear?
5. How does the formula change if you know the first and last terms but not the number of terms?
6. What is the significance of arithmetic progressions in mathematics?
7. How would the sum change if the common difference was different?
8. Can you derive the sum formula for an arithmetic progression?

**Tip:** Remember that the sum of an arithmetic series increases quadratically with the number of terms, even if the common difference is constant.

Learn how to find the sum of an arithmetic progression (AP) where the first term is 2, the last term is 152, and the common difference is 2. Detailed step-by-step solution using the arithmetic progression sum formula. Suitable for high school students studying arithmetic sequences.

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