Understanding the Arithmetic Sequence and Finding the Sum of Its First 15 Terms

When dealing with arithmetic sequences, we often need to find specific terms or the sum of the first n terms. This article will focus on finding the sum of the first 15 terms of an arithmetic sequence where the sixth term is -9 and the tenth term is -15. Let's explore the steps involved in solving such a problem.

Identifying the Common Difference d

To start, we know that the sixth term ((t_6)) is -9 and the tenth term ((t_{10})) is -15. We can use the formula for the nth term of an arithmetic sequence: (t_n a (n-1)d), where a is the first term and d is the common difference.

Calculating the Common Difference

First, we can set up the following equations based on the given terms:

[t_6 a 5d -9]

[t_{10} a 9d -15]

To find the common difference d, we subtract the first equation from the second:

[ (a 9d) - (a 5d) -15 - (-9) Rightarrow 4d -6 Rightarrow d -frac{3}{2}]

Determine the First Term a

Now that we have the value of d, we can substitute it back into one of the equations to find the first term a. We'll use the equation for the sixth term again:

[a 5d -9 Rightarrow a 5 left(-frac{3}{2}right) -9 Rightarrow a - frac{15}{2} -9 Rightarrow a -9 frac{15}{2} -frac{18}{2} frac{15}{2} -frac{3}{2}]

Calculating the Sum of the First 15 Terms

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

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

For the first 15 terms ((n 15)), we can substitute the values of a and d into the formula:

[S_{15} frac{15}{2} left(2 left(-frac{3}{2}right) 14 left(-frac{3}{2}right)right) frac{15}{2} left(-3 - 21right) frac{15}{2} times -24 -15 times 12 -180]

This calculation shows that the sum of the first 15 terms of the given arithmetic sequence is -180.

Verification and Further Exploration

To verify this, let's review the sequence step-by-step. Starting from the first term a -frac{3}{2} and the common difference d -frac{3}{2}, we can generate the sequence as follows:

(-frac{3}{2}, -3, -frac{9}{2}, -6, -frac{15}{2}, -9, ldots) (-15, -frac{21}{2}, -12, -frac{27}{2}, -15, -frac{33}{2}, ldots)

The pattern holds, and we can see that the sum of the first 15 terms is indeed -180.

Key Points: The common difference d is -1.5. The first term a is -1.5. The sum of the first 15 terms is -180.

This problem reinforces the understanding of arithmetic sequences, their terms, and the sum of such sequences. It's a valuable exercise for students and educators to grasp the concept and its practical implications.

---

If you find this explanation helpful, feel free to explore more problems or refer to educational resources for further practice. Happy learning!