Problem 107 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the three numbers constituting a G.P. if it is known that the sum of the numbers is equal to 26 and that when 1,6 and 3 are added to them respectively, the new numbers are obtained which from an A.P.

Step-by-Step Solution

Answer

Based on the solution of the above equations, the three numbers which form the geometric progression under given conditions will be obtained.

Step 1: Set Up the Problem

Let the three numbers be \(a/r\), \(a\) and \(ar\) as these are in geometric progression. Now from the question it is known that the sum of these three numbers is 26. So we get the equation \(a/r + a + ar = 26\)

Step 2: Applying conditions to form equations

Next, the question states when 1, 6 and 3 are added to the numbers respectively, they form an arithmetic series. This gives us two equations based on properties of an arithmetic progression: (1) The sum of first and third term is equal to twice the second term, and (2) The difference of the second term and the first term is equal to the difference of the third term and the second term. This translates into the following equations: (1) \(a/r + 1 + ar + 3 = 2(a + 6)\) and (2) \( a+6 - (a/r+1) = (ar+3) - (a+6)\)

Step 3: Solve for variables

Now solve these equations to find the values of \(a\) and \(r\). First transform first equation into the form that may be solved using factorization and solve it. Then substitute the obtained values of \(r\) in the equation \(ar+3 = a+6\), this will give the values for \(a\). After that substitute these values of \(r\) and \(a\) in the equation \(a/r + a + ar = 26\). This should provide us with solutions for \(a\) and \(r\) that meet all conditions

Step 4: Answer the Problem

Now substitute \(a\) and \(r\) into the original definitions to solve for the actual numbers. So, the numbers are \(a/r\), \(a\) and \(ar\) respectively which constitute our geometric progression.

Key Concepts

Geometric Progression

A geometric progression, or GP, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio. Understanding the concept of a common ratio is crucial:

• If the common ratio is greater than 1, the sequence will grow exponentially.
• If the common ratio is between 0 and 1, the sequence will decrease exponentially.
• If the common ratio is negative, the sequence terms will alternate between positive and negative values.

In the problem statement, we are given three numbers in GP: \(\frac{a}{r}\), \(a\), and \(ar\). This means:

• \(\frac{a}{r}\) is the first term.
• \(a\) is the second term, found by multiplying the first term by \(r\).
• \(ar\) is the third term, found by multiplying the second term by \(r\) again.

These numbers together satisfy the condition of the sum being 26. Recognizing and setting up sequences this way is key to solving the equation and finding specific values in a geometric sequence.

Arithmetic Progression

An arithmetic progression, or AP, is a sequence of numbers in which the difference between consecutive terms is constant. This constant is known as the common difference. The simplicity of this concept comes from how predictable the sequence is. If you know the first term and the common difference, you can construct the entire sequence.

• The sequence 2, 4, 6, 8 is an AP with a common difference of 2.
• The sequence 10, 7, 4, 1 is an AP with a common difference of -3.

In this exercise, after adding a set number to each of the geometric progression terms, the new sequence forms an AP. The conditions to form an arithmetic progression are:

• The sum of the first and third terms should be twice the second term: \( (a/r + 1) + (ar + 3) = 2(a + 6) \).
• The difference between the second and first term is equal to the difference between the third and second term: \( (a+6) - (a/r+1) = (ar+3) - (a+6) \).

These conditions provide equations that enable us to solve for the unknowns in the problem using properties of AP.

Simultaneous Equations

Simultaneous equations are a set of equations containing multiple variables. The solutions are the values of the variables that satisfy all equations in the set simultaneously. These are solved using methods like substitution, elimination, or matrix operations. This practice is important in a variety of fields such as physics, engineering, and economics.In the exercise, simultaneous equations are formed based on the conditions of both the geometric and arithmetic progressions:

• The first equation, \(\frac{a}{r} + a + ar = 26\), represents the sum condition of the GP.
• The second set of equations, \((a/r + 1) + (ar + 3) = 2(a + 6)\) and \((a+6) - (a/r+1) = (ar+3) - (a+6)\), characterize the resultant AP.

To solve simultaneous equations, one could start with simpler equations to express one variable in terms of another, then substitute back into other equations. For example, solving the second set for \(a\) and \(r\), these variables can then be substituted into the GP sum equation. Mastering this method is crucial for developing strong problem-solving abilities in algebra.