MathGram - for both Mathematics and Programming

Question

Given the first term $ a = 5 $ and the common difference $ d = 3 $, find the sum of the first 50 terms of the arithmetic progression.

Step-by-Step Solution

To solve for the sum of the first $ n $ terms of an arithmetic progression, we use the formula:

$$ S_n = \frac{n}{2} \times (2a + (n - 1)d) $$

Where:

$ S_n $ is the sum of the first $ n $ terms
$ n $ is the number of terms
$ a $ is the first term
$ d $ is the common difference

Given:

$ a = 5 $
$ d = 3 $
$ n = 50 $

Substitute these values into the formula:

$$ S_{50} = \frac{50}{2} \times (2 \times 5 + (50 - 1) \times 3) $$

Simplify inside the parentheses:

$$ S_{50} = 25 \times (10 + 147) $$

$$ S_{50} = 25 \times 157 $$

Multiply to get the final sum:

$$ S_{50} = 3925 $$

So, the sum of the first 50 terms of the arithmetic progression is 3925.

Solving the Problem with Python

Now, let's solve this problem using Python. We will define a function to calculate the sum of the first $ n $ terms of an arithmetic progression.

```python
def sum_of_ap(n, a, d):
    # Calculate the sum using the formula
    sum_n = n / 2 * (2 * a + (n - 1) * d)
    return sum_n

# Given values
a = 5
d = 3
n = 50

# Calculate the sum
sum_50_terms = sum_of_ap(n, a, d)
print(f"The sum of the first {n} terms of the arithmetic progression is {sum_50_terms}")

Explanation of the Python Code

Define the Function:
We define a function sum_of_ap that takes the number of terms $ n $, the first term $ a $, and the common difference $ d $ as inputs. The function calculates the sum using the arithmetic progression sum formula.
Given Values:
We assign the given values of $ a $, $ d $, and $ n $.
Calculate and Print the Sum:
We call the sum_of_ap function with the given values and print the result.

By following these steps, we've solved a challenging arithmetic progression problem both manually and programmatically. Stay tuned for more math challenges and solutions on SkoolTest!