Understanding and finding the arithmetic sequence is a fundamental skill in algebra. In this article, we will guide you through a detailed step-by-step process to find an arithmetic sequence given two specific terms of the sequence. We will focus on the practical application of this concept and ensure that the information is structured in a way that aligns with Google's SEO standards.

Introduction to Arithmetic Sequences

Arithmetic sequences are sequences of numbers in which each term increases or decreases by a constant value, known as the common difference. For instance, in the sequence 3, 7, 11, 15, the common difference (d) is 4. Each term is obtained by adding the common difference to the previous term.

Problem Statement

The problem at hand is to find the arithmetic sequence given that the fifth term (U5) is -2 and the twelfth term (U12) is -12.5. We will use the general formula for the nth term of an arithmetic sequence: U_n a (n-1)d, where a is the first term, d is the common difference, and n is the term number.

Solving the Problem

Let's start by setting up the problem using the given terms. We know:

U5 -2 U12 -12.5

Using the general formula, we can express these terms as:

U5 a 4d U12 a 11d

From equation (1) and (2), we have:

Equation 1: -2 a 4d

Equation 2: -12.5 a 11d

Step 1: Eliminate the Variable a

To eliminate the variable 'a', we can subtract equation 1 from equation 2:

" "-12.5 a 11d" " " "(-2) a 4d" "

Subtracting (1) from (2) gives us:

-12.5 - (-2) (a 11d) - (a 4d)

Simplifying this, we get:

-10.5 7d

Solving for d, we find:

d -1.5

Step 2: Find the First Term a

Now that we know the common difference d, we can substitute d back into either equation (1) or (2) to find the first term a. Let's use equation (1):

-2 a 4d

Substituting d -1.5 into the equation:

-2 a 4(-1.5)

Now, solving for a gives us:

-2 a - 6

Adding 6 to both sides, we get:

a 4

Step 3: Construct the Arithmetic Sequence

Now that we have a 4 and d -1.5, we can find each term of the sequence by adding d to the previous term. Starting with U1 a 4:

U1 4 U2 4 - 1.5 2.5 U3 2.5 - 1.5 1 U4 1 - 1.5 -0.5 U5 -0.5 - 1.5 -2

Thus, the arithmetic sequence is:

4, 2.5, 1, -0.5, -2, ...

Conclusion

By following the steps outlined above, we can find the arithmetic sequence when given specific terms. The key steps involve setting up equations based on the given terms, eliminating the first term to find the common difference, and then using the common difference to find the rest of the terms in the sequence. Understanding these steps not only helps in solving similar problems but also deepens the understanding of arithmetic sequences.

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