Calculating the Sum of the First 10 Terms of an Arithmetic Sequence

An arithmetic sequence is a series of numbers where each term after the first is found by adding a constant, known as the common difference, to the previous term. In this article, we will explore how to find the sum of the first 10 terms of an arithmetic sequence given the first term and the common difference.

Understanding the Arithmetic Sequence

Let's consider an arithmetic sequence where the first term, denoted as a1, is 5, and the common difference, denoted as d, is -3. Using these values, we can determine the first 10 terms of the sequence.

First 10 Terms of the Sequence

ta1 5 ta2 2 ta3 -1 ta4 -4 ta5 -7 ta6 -10 ta7 -13 ta8 -16 ta9 -19 ta10 -22

The sum of the first 10 terms can be calculated using several methods. Let's explore these methods one by one.

Using Direct Addition

We can find the sum of the first 10 terms by directly adding the terms:

t5 2 (-1) (-4) (-7) (-10) (-13) (-16) (-19) (-22) -85

The sum of the first 10 terms is indeed -85.

Using the Sum Formula

Alternatively, we can use the formula for the sum of the first n terms of an arithmetic sequence:

Snn2[2au22121d]

Where:

tSn tThe sum of the first n terms ta tThe first term td tThe common difference tn tThe number of terms

Substituting n 10, a 5, and d -3 into the formula:

S10102[2×5u22121×3]5×10u221235×37185

The sum of the first 10 terms is 185, but this result is incorrect. Let's correct the formula and recalculate.

Correcting the Formula and Recalculating

The correct formula to use is:

S10102[2a 9d]5×2×5 9×?35×10u2212275×?17u221285

The correct sum of the first 10 terms is -85.

Conclusion

By understanding the formula for the sum of an arithmetic sequence and plugging in the appropriate values, we can easily find the sum of the first 10 terms. In this case, the sum of the first 10 terms of an arithmetic sequence where the first term is 5 and the common difference is -3 is -85.