Question

Three alternate terms of an arithmetic progression are \(x+y, x-y\) and \(2 x+3 y\), then \(x=\) (1) - y (2) \(-2 \mathrm{y}\) (3) \(-4 y\) (4) -6y

Short answer

Question: Given three alternate terms of an arithmetic progression: \(x+y, x-y\) and \(2x+3y\), find the value of x in terms of y. Answer: \(x = -6y\)

Step-by-step solution

Write the given arithmetic progression terms

We are given three alternate terms of an arithmetic progression: \(x+y, x-y\) and \(2x+3y\). Let's call them term1, term2, and term3. term1 = \(x+y\) term2 = \(x-y\) term3 = \(2x+3y\)

Determine the common difference

As we know that the common difference between any two consecutive terms is always constant, let's find this common difference (d). As term2 and term1 are alternate terms, we can find the common difference as: d = term2 - term1 = \((x-y) - (x+y)\) = \(-2y\) Similarly, the common difference between term3 and term2 is: d = term3 - term2 = \((2x+3y) - (x-y)\) = \((x+4y)\)

Form an equation and solve

We now have two expressions for the common difference: \(-2y\) = \((x+4y)\) Now, solve for x using algebra techniques: \((x+4y) = -2y\) \(x = -6y\) So, the value of x is -6y. The correct answer is (4) -6y.

Key concepts

Arithmetic Sequence

An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference. In an arithmetic sequence, any term after the first can be calculated by adding the common difference to the previous term. For example, if the first term of an arithmetic sequence is 5 and the common difference is 3, the sequence would be 5, 8, 11, 14, and so on. Important properties of an arithmetic sequence include its ability to be represented by a linear algebraic expression, and that the sum of a finite number of terms can be quickly calculated using a simple formula.

Common Difference

The common difference in an arithmetic sequence is the consistent interval or difference between neighboring terms. It is denoted by the symbol 'd'. To find the common difference, you subtract a term from the term that follows it. This difference remains the same throughout the entire sequence. For example, if you are given an arithmetic sequence where the second term is 7 and the third term is 12, the common difference would be calculated as follows: D = term3 - term2, which gives us a common difference of 5. The common difference is crucial because it defines the unique characteristic of the arithmetic sequence—that each term increases by a fixed amount.

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operations (like addition, subtraction, multiplication, and division). It represents a value that can change depending on the variables involved. Algebraic expressions are used to describe patterns, relations, and changes in quantities. They become exceptionally useful when we deal with sequences, as they allow us to express general terms using variable representations of any element in that sequence based on its position. For example, in an arithmetic sequence, the n-th term can be expressed as \( a_n = a_1 + (n - 1)d \) where \( a_1 \) is the first term and \( d \) is the common difference.

Solving Equations

Solving equations is the process of finding the value of the variable that makes the equation true. Equations can be simple, with one variable, or very complex, involving multiple variables and operations. The basic principle is to isolate the variable on one side of the equation while keeping the equation balanced. For arithmetic sequence problems, solving equations often involves finding the value of a term in the sequence or the common difference. By manipulating the algebraic expressions that represent the terms, we can solve for the unknowns. In the example from the exercise, equating two expressions representing the common difference and solving for \( x \) gives us the answer to a term in the sequence.