How to Find the Sum of the First 15 Terms in an Arithmetic Sequence

The arithmetic sequence given is 31, 40, 49, and 58. This sequence is characterized by a common difference of 9 between each term. To find the sum of the first 15 terms, we can use both mathematical formulas and Python code.

Multiplying Formula Approach

To find the sum of the first 15 terms of an arithmetic sequence, we can use the formula for the sum of an arithmetic progression (AP), which is given by:

S_n frac{n}{2} [2a (n-1)d]

Here:

a 31 (first term) d 9 (common difference) n 15 (number of terms)

Substituting these values into the formula, we get:

S_{15} frac{15}{2} [2 cdot 31 (15-1) cdot 9]

Calculate inside the parentheses:

S_{15} frac{15}{2} [62 14 cdot 9]

S_{15} frac{15}{2} [62 126]

S_{15} frac{15}{2} cdot 188 15 cdot 94 1410

Thus, the sum of the first 15 terms is 1410.

Reversing and AddingVertically

As an alternative approach, we can also find the 15th term of the sequence by iteratively adding the common difference:

31 9 40 40 9 49 49 9 58 58 9 67 67 9 76 76 9 85 85 9 94 94 9 103 103 9 112 112 9 121 121 9 130 130 9 139 139 9 148 148 9 157

Next, we write the sequence and its reverse:

S 31, 40, 49, 58, 67, 76, 85, 94, 103, 112, 121, 130, 139, 148, 157

S 157, 148, 139, 130, 121, 112, 103, 94, 85, 76, 67, 58, 49, 40, 31

Adding these vertically, we get:

2S 188, 188, 188, 188, 188, 188, 188, 188, 188, 188, 188, 188, 188, 188, 188

2S 188 times 15 2820

S frac{2820}{2} 1410

Using Python for Arithmetic Progressions

Python provides a simple and efficient way to calculate the sum of an arithmetic sequence. Here is the Python code to find the sum of the first 15 terms of the sequence:

sum([31 i * 9 for i in range(15)]) 1410

Let's break down the Python code:

sum([31 i * 9 for i in range(15)]): This statement generates the arithmetic sequence and then calculates its sum. 31 i * 9: This expression determines the ith term of the sequence by starting from 31 and adding 9 times i to it. range(15): This generates a sequence of numbers from 0 to 14, which we use to generate the terms of the sequence.

By using this approach, you can easily find the sum of any arithmetic sequence using Python.

Conclusion

Whether you prefer using the traditional mathematical formula or leveraging the power of programming with Python, finding the sum of the first 15 terms of the given arithmetic sequence is straightforward. The sum is 1410, as confirmed by both methods discussed.

Next Steps

Expand your understanding of arithmetic sequences by trying out similar problems or diving deeper into the properties of arithmetic progressions. Practice can improve your skills and provide a solid foundation for more complex mathematical concepts.