Finding the Value of p in an Arithmetic Sequence

In this article, we will explore the problem of determining the value of p in an arithmetic sequence where the terms are 11, p, q, 43/2. We'll discuss the properties of arithmetic sequences, how to find the common difference, and solve the algebraic equations to find the desired value.

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference (d). In other words, for any arithmetic sequence a, a d, a 2d, ..., a (n-1)d, the difference between any two successive terms is always equal to d.

Setting Up the Problem

We are given the arithmetic sequence 11, p, q, 43/2. The sequence should satisfy the properties of an arithmetic sequence, meaning the difference between consecutive terms remains constant.

Solving for p and q

We start by setting up the equations based on the common difference. For an arithmetic sequence, the common difference can be expressed as:

d p - 11 d q - p d 43/2 - q

Equating these expressions for d, we get:

p - 11 q - p q - p 43/2 - q

From the first equation, we can rearrange to solve for q in terms of p:

2p q 11

q 2p - 11.

The second equation simplifies to:

2q 43/2 p

q 43/4 p/2

Now, we can use the fact that these two equations represent the same value for q to solve for p. First, let's set the two expressions for q equal to each other:

2p - 11 43/4 p/2

Multiplying through by 4 to clear the fractions, we get:

8p - 44 43 2p

Simplifying further:

6p 87

p 87/6 29/2 14.5

Verifying the Solution

Substituting p 29/2 back into one of the equations for q, we find:

q 2(29/2) - 11 29 - 11 18

The sequence thus becomes:

11 29/2 14.5 18 43/2 21.5

Checking the common difference, we find:

14.5 - 11 3.5 18 - 14.5 3.5 21.5 - 18 3.5

The common difference is indeed 3.5, confirming our solution is correct.

Conclusion

We have successfully found the value of p to be 29/2 or 14.5 in the given arithmetic sequence. This problem illustrates the use of algebraic equations and the properties of arithmetic sequences to determine unknown values.