Question

If \(a, b, c, d\) are in GP, then \((b+c)^{2}=\) _______. (1) \((b+d)(a+d)\) (2) \((a+d)(c+d)\) (3) \((a+b)(c+d)\) (4) \((a+c)(b+d)\)

Short answer

Answer: (b+c)^2 = (a+c)(b+d)

Step-by-step solution

Determine the common ratio

Since a, b, c, and d are in gp, there exists a common ratio "r" such that b=ar, c=br, and d=cr. We can write the expressions for these terms as follows: b=ar c=ar^2 d=ar^3 Now we can use these expressions in the given equation.

Substitute the expressions for b, c, and d

Let's substitute the expressions for b and c in the equation (b+c)^2 and simplify the expression: \((ar+ar^2)^2\)

Simplify the expression

Now, we'll simplify the given expression by expanding it: \((ar+ar^2)^2 = (ar^2 + 2(ar)(ar^2) + (ar^2)^2)\) Which simplifies to: \(= a^2r^2 + 2a^2r^3 + a^2r^4\) Now, let's factor the expression: \(= a^2r^2(1 + 2r + r^2)\)

Compare with the given options

We have reached the expression \((b+c)^2 = a^2r^2(1 + 2r + r^2)\). Now we need to compare it with the given options to find the correct answer. Let's replace r with their respective values. From our earlier analysis, we can replace r by b/a, r^2 by c/a, and r^3 by d/a. So we have: \(b+c = a(b/a)+ar^2\) \(b+c = b+ar^2\) Now, compare this expression with the given options: (1) \((b+d)(a+d)\) does not match (2) \((a+d)(c+d)\) does not match (3) \((a+b)(c+d)\) does not match (4) \((a+c)(b+d) = (a+ar^2)(ar + ar^3) = (a + c)(b + d)\) Thus, option (4) is correct. Therefore, \((b+c)^2 = (a+c)(b+d)\).

Key concepts

Geometric Progression (GP)

In mathematics, a Geometric Progression (GP) or Geometric Sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in the sequence 2, 6, 18, 54, each term is obtained by multiplying the previous one by 3, making this a GP with a common ratio of 3.

A GP series can be represented algebraically as:

• First term (a): the first number in the sequence.
• Common ratio (r): the constant factor between consecutive terms.
• nth term (a_n): a * r^{(n-1)}, which calculates any term based on the first term and the common ratio.
• Sum of the first n terms (S_n): a(1 - r^n) / (1 - r) for

Understanding the concept of GP is essential for solving many problems in algebra, as it frequently appears in problems involving exponential growth or decay, financial computations, and in the analysis of sequences.

Common Ratio in GP

The common ratio in a GP is the factor by which every term of the sequence is multiplied to get the next term. It is denoted by the letter r. The common ratio can be found by dividing any term in the GP by the previous term, assuming the sequence has at least two terms. In the context of our problem, having a GP with terms a, b, c, d, we obtain the common ratio r as follows: b / a, c / b, or d / c, which all should give the same number because the ratio is constant throughout the GP.

The common ratio is crucial in determining various properties of the sequence, including convergence, where if |r| < 1, the GP is convergent. Otherwise, if |r| > 1, the sequence diverges, meaning the terms grow without bound as n increases.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. They are symbols put together in a meaningful way to represent numbers, relations amongst numbers, or operations applied to unknowns or other expressions. In our exercise, the expression (b+c)^2 is an algebraic expression that we need to simplify and match to given options.

To work with algebraic expressions, especially in the context of a GP, we often replace the terms of the progression with the expression that defines them in terms of the first term and the common ratio. This helps in simplifying and manipulating the expressions to reach a desired form, as seen in the step-by-step solution provided in the original problem.

Problem Solving in Mathematics

Problem solving in mathematics involves understanding the problem, devising a plan, carrying out the plan, and looking back to check the results. These are classic steps outlined by the mathematician George Polya. Applying these steps to solve problems with geometric progressions generally involves:

• Identifying that a sequence is indeed a GP.
• Determining the common ratio and expressing the terms of the GP in terms of the first term and this ratio.
• Using algebraic skills to manipulate expressions and equations involving terms of the GP.
• Comparing the derived expressions to the given conditions or answer options.

As seen in the provided step-by-step solution, the problem was approached by recognizing the GP, finding the common ratio, substituting terms of the GP into the given algebraic expression, and simplifying it to find the correct match amongst the options. This structured approach exemplifies effective problem solving in mathematics.