Problem 73 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

The sum of first \(p\) terms of an A.P. is \(q\) and the sum of the first \(q\) terms is \(p\). Find the sum of the first \(p+q\) terms.

Step-by-Step Solution

Answer

The sum of the first \(p+q\) terms can be calculated by solving the system of equations to find the values of \(a\) and \(d\), followed by substituting these obtained values into \(S_{p+q} = (p+q)/2 * (2a + (p+q-1)d)\). The specific result will depend on the given \(p\) and \(q\) in the question.

Step 1: Determine the first terms and their sum

From the given information, we can derive that the sum of the first \(p\) terms is \(q\), which can be written as \(S_p=q\) or \(p/2 * (2a + (p-1)d) = q\), where \(a\) is the first term and \(d\) is the common difference. From this, we see the first relationship between \(a, d, p,\) and \(q\).

Step 2: Use the sum formula for the second set of terms

Similarly, we can imply that the sum of the first \(q\) terms is \(p\), which can be shown as \(S_q=p\) or \(q/2 * (2a + (q-1)d) = p\). This creates another mathematical relationship we can use.

Step 3: Solve the system of equations

Now, we have two equations and two unknowns. We can use the method of substitution or elimination to solve this system of equations to find the values of \(a\) and \(d\).

Step 4: Use the sum formula for the \(p+q\) terms

Once we know \(a\) and \(d\), we can find the sum of \(p+q\) terms using the formula: \(S_{p+q} = (p+q)/2 * (2a + (p+q-1)d)\). This is the final step to calculate the sum of the first \(p+q\) terms.

Key Concepts

Sum of Series

The concept of the sum of a series in an Arithmetic Progression (A.P.) is crucial for solving the given problem. An arithmetic series is basically an ordered list of numbers where each term after the first is obtained by adding a constant, known as the "common difference." The sum of the series for the first few terms is given by a specific formula. For an A.P. with 'n' terms, the sum formula is:

• \( S_n = \frac{n}{2} \left(2a + (n-1)d \right) \)

Here:

• 'n' is the number of terms.
• 'a' is the first term.
• 'd' is the common difference.

Understanding and applying this formula allows one to express relationships and solve for unknowns within the arithmetic series. The problem requires using the formula twice for two different sets of terms to draw conclusions.

System of Equations

A system of equations is a set of two or more equations with the same variables. The goal is to find the values of these variables that satisfy all the given equations simultaneously.
In our problem, we derive two equations based on the sum of series:

• Equation 1: \( \frac{p}{2} (2a + (p-1)d) = q \)
• Equation 2: \( \frac{q}{2} (2a + (q-1)d) = p \)

Here, the variables "a" and "d" are unknown, and "p" and "q" are given values. Solving this system can be done using:

• Substitution Method: Solve one equation for one variable and substitute it into the other equation.
• Elimination Method: Adjust the equations to eliminate one variable by adding or subtracting them from one another.

Finding both "a" and "d" gives us the first step towards finding the sum of terms later on.

Common Difference

In an Arithmetic Progression (A.P.), the common difference is a key element. It represents the constant amount that you add to each term in a sequence to get the next term.

• For example, in the A.P. 2, 4, 6, 8, the common difference (d) is 2.

The common difference is crucial for constructing the series and is part of the sum of series formula:

• \( S_n = \frac{n}{2} (2a + (n-1)d) \)

It appears in the part of the formula \((n-1)d)\) and directly influences the sum and behavior of the series.
In problems like these, finding "d" helps to build the series and find other unknowns like the first term "a." This then aids in calculating sums for larger sets of terms.

First Term

The first term, often denoted as "a," is the starting number of an arithmetic sequence. It's the cornerstone of the entire progression.

• For the A.P. with terms 3, 7, 11, 15, "a" would be 3.

"a" is used in the sum of series formula:

• \( S_n = \frac{n}{2} (2a + (n-1)d) \)

Knowing "a" is essential for forming the sequence of terms and calculating their sum. In the given problem, "a" is one of the unknown quantities that is found using the system of equations derived from the problem's conditions.
After determining "a" alongside "d," you can effectively apply the sum formula to find the sum of the first \(p+q\) terms, completing the solution.