Arithmetic Sequences: Calculating the Sum and Terms of an AP

Arithmetic sequences (AP) are a fundamental concept in mathematics that have wide-ranging applications, from financial planning to engineering. If you are familiar with the first term and the common difference of an arithmetic sequence, you can determine any term of the sequence and the sum of the first n terms. This article will guide you through the process of finding the first 28 terms of an AP where the first term (a) is 3 and the common difference (d) is 4. We will use both formulas and steps to calculate the sum of the first 28 terms and some specific terms within this sequence.

Calculating the Sum of the First 28 Terms

The formula to find the sum of the first n terms of an arithmetic sequence is given by:

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

Here, n is the number of terms, a is the first term, and d is the common difference. Let us apply this formula to find the sum of the first 28 terms.

Given a 3 and d in the values into the formula:

Sn  frac{28}{2} [2(3)   (28-1)4]

Calculate inside the brackets:

Sn  14 [6   27 times 4]

Further simplify:

Sn  14 [6   108]

To get the final answer:

Sn  14 times 114  1596

Thus, the sum of the first 28 terms is 1596.

Calculating Specific Terms of the AP

Let's explore the first 28 terms of the arithmetic sequence more closely and find its general term and sum using the provided values of the first term and common difference.

For the 11th term:

The formula to find the nth term of an arithmetic sequence is given by:

T_n a (n-1)d

For the 11th term (T_11):

T_11 3 (11-1) times 4

T_11 3 10 times 4

T_11 3 40 43

Therefore, the 11th term of the sequence is 43.

General Term and Summation:

The general term of the sequence can be expressed as:

T_n 4n - 1

The sum of the first 28 terms can be found by plugging the values into the sum formula:

S_{28} frac{28}{2} [2(3) 27 times 4]

S_{28} 14 [6 108]

S_{28} 14 times 114 1596

This gives us the sum of the first 28 terms as 1596.

General Formulas and Conclusion

The arithmetic sequence can be written as:

3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, …

The nth term of this sequence can be given by the formula:

T_n 4n - 1

The sum of the first n terms of this sequence can be given by:

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

For n 28, we have derived that the sum is:

S_{28} 1596

In conclusion, understanding and applying these formulas to find specific terms and the sum of an arithmetic sequence are crucial skills in mathematics. Whether you are dealing with financial investments, engineering calculations, or other real-world applications, these concepts are a cornerstone of problem-solving.