Problem 100 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

In a set of four numbers the first three are in G.P. and the last three in A.P. with common difference \(6 .\) If the first number is the same as the fourth, find the four numbers.

Step-by-Step Solution

Answer

The four numbers that satisfy the given conditions are 6, 6, 6, 12.

Step 1: Understanding Arithmetic and Geometric Progressions

Geometric Progression (G.P) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number. Let's consider the first three numbers of our sequence as \(a\), \(ar\) and \(ar^2\), where \(a\) is the first term and \(r\) is the common ratio of the G.P. An Arithmetic Progression (A.P.) is a sequence of numbers in which the difference between any two successive members is a constant. The last three numbers are considered as \(ar\), \(ar^2\) and \(ar^2+6\) for this sequence, where \(ar\) is the first term, and the common difference is 6.

Step 2: Formulate Equation from conditions

The conditions state that the first term is equal to the fourth, which means \(a = ar^2+6\). Through solving this equation we can find the value of \(r\). As \(a\) is not zero, we can divide the entire equation by \(a\) and get \(r^2+6/a=1\). Thus, \(r^2=1-6/a\)

Step 3: Formulate second Equation from AP

For A.P., we have the condition that the common difference is 6, meaning \(ar^2+6-ar=6\). The \(ar\) simplifies to \(ar^2-ar=6\). When dividing the entire equation by \(a\) we have \(r^2-r=6/a\). Combining this information with the equation from Step 2: \(r^2=1-6/a\) equals \(r^2=6/a+r\)

Step 4: Solve for a

Rearrange the equation from Step 3 to be in terms of \(a\), giving \[a=6/(1-2r)\]. We know \(a\) cannot be zero and \(r\) cannot be \(0.5\) (since the denominator cannot be zero). Thus we find that \(a=6\) and \(r=1\)

Step 5: Finding the numbers

Substitute the value of \(a\) and \(r\) into the progression sequences i.e. \(a\), \(ar\), \(ar^2\) and \(ar^2+6\). This gives the four numbers as 6, 6, 6 and 12.

Key Concepts

Geometric Progression (G.P.)

A Geometric Progression (G.P.) is a sequence of numbers where each term after the first is the product of the previous term and a constant called the common ratio. For example, in the progression 2, 4, 8, 16, each number is multiplied by 2, making 2 the common ratio. If the first term is denoted by \(a\) and the common ratio by \(r\), the terms of a G.P. can be represented as \(a, ar, ar^2, ar^3, \ldots\).
Understanding G.P. allows us to solve complex problems involving exponential growth or decay. It's essential to determine both the first term and the common ratio to find any number in the sequence or the sum of these numbers. In our problem, knowing the sequence starts as \(a\), \(ar\), and \(ar^2\), and how it evolves helps us solve the exercise.

Arithmetic Progression (A.P.)

An Arithmetic Progression (A.P.) is a number sequence where each term after the first is obtained by adding a constant, called the common difference, to the previous term. Unlike G.P., this common difference remains consistent throughout the progression. A simple A.P example is 3, 6, 9, 12, where the common difference is 3.For an A.P., if we denote the first term by \(b\) (like \(ar\) in our problem) and the common difference by \(d\), the sequence becomes \(b, b+d, b+2d, b+3d, \ldots\). In our exercise, recognizing that the last three numbers form an A.P. with a common difference of 6 helps derive the relationships needed to find specific terms, crucial in forming and solving the equations.

Common difference

The common difference in an Arithmetic Progression (A.P.) is the fixed amount added to each term to reach the next term. It is a key feature that defines an A.P. In our problem, the common difference is given as 6.
This common difference is crucial for understanding how the sequence progresses and for setting up equations that relate different terms. Specifically, knowing that the common difference is consistent across the sequence makes it possible to calculate unknowns in the progression. In our exercise, it allows us to set up equations such as \(b+d=ar^2+6\), simplifying the problem-solving process.