What is the Sum of the First 30 Terms in an Arithmetic Sequence?

The arithmetic sequence provided is as follows: 6, 13, 20, 27, 34. To solve this problem, we will walk through the steps to determine the sum of the first 30 terms using the formulas and key concepts associated with arithmetic sequences.

Identifying the First Term and Common Difference

Let's start by identifying the first term and the common difference of the sequence:

First term a: The first number in the sequence, which is 6. Common difference d: The fixed difference between any two consecutive terms. In this sequence, the common difference can be calculated by subtracting the first term from the second term: d 13 - 6 7.

Finding the 30th Term

Next, we will find the 30th term using the formula for the nth term of an arithmetic sequence:

a_n a (n - 1) × d

For the 30th term a_{30} using n 30:

a_{30} 6 (30 - 1) × 7
a_{30} 6 29 × 7
a_{30} 6 203
a_{30} 209

Calculating the Sum of the First 30 Terms

Now, we will calculate the sum of the first 30 terms of this arithmetic sequence using the formula for the sum of the first n terms of an arithmetic sequence:

S_n frac{n}{2} × (a a_n)

For n 30, a 6, a_{30} 209:

S_{30} frac{30}{2} × (6 209)

S_{30} 15 × 215

S_{30} 3225

Conclusion

In summary, the sum of the first 30 terms of the arithmetic sequence 6, 13, 20, 27, 34 is 3225. This problem demonstrates the application of arithmetic sequence formulas to solve for a specific term and the sum of a series of terms.

Additional Tips for Solving Arithmetic Sequence Problems

Simplify the formula: As shown above, the formula for the 30th term and the sum of the first 30 terms can be simplified to make calculations easier. Identify the key information: Always ensure you identify and label the first term and common difference before proceeding with the calculations. Practice regularly: Regular practice with various arithmetic sequence problems will improve your problem-solving skills and help you tackle more complex sequences.

By mastering the steps and understanding the underlying concepts of arithmetic sequences, you can confidently solve similar problems and apply these skills to a wide range of mathematical and real-world scenarios.