Problem 63 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

The first and last term of an A.P. are \(a\) and \(l\) respectively. If \(S\) be the sum of all the term of the A.P., show that the common difference is \(\frac{l^{2}-a^{2}}{2 S-(l+a)}\).

Step-by-Step Solution

Answer

The common difference of the arithmetic progression with first term \(a\), last term \(l\), and sum \(S\) can be shown to be \(\frac{l^2 - a^2}{2S - (l+a)}\).

Step 1: Sum of Arithmetic Progression

The sum of the terms in an arithmetic progression can be calculated using the formula: \[S = \frac{n}{2}(a + l)\] Where \(S\) is the sum of the terms in the progression, \(n\) is the number of terms in the progression, \(a\) is the first term, and \(l\) is the last term.

Step 2: Nth Term of Arithmetic Progression

The nth term of an arithmetic progression is given by the formula: \[T_n = a + (n-1) d\] Where \(T_n\) is the nth term, \(a\) is the first term, \(n\) is the number of terms, and \(d\) is the common difference.

Step 3: Substitute the Nth Term Formula into the Sum Formula

In this step, we will substitute T_n (where T_n = l) for the last term in the sum formula: \[S = \frac{n}{2} (a + (a + (n - 1) d))\]

Step 4: Simplify the Equation

Simplify the above equation to get: \[S = \frac{n}{2} (2a + (n - 1) d)\]

Step 5: Solve for the Common Difference (d)

Rearrange the equation to solve for the common difference: \[d = \frac{2S - n(2a)}{n(n - 1)}\]

Step 6: Substitute l for the Nth Term

Now, let's substitute \(l\) for \(T_n = a + (n - 1) d\) in the nth term formula: \[l = a + (n - 1) d\]

Step 7: Solve for n in terms of a, l, and d

Rearrange the equation from step 6 to solve for n: \[l - a = (n - 1) d\] \[n = \frac{l - a}{d} + 1\]

Step 8: Substitute the new value of n in the equation from step 5

Substitute the value of n from step 7 into the equation from step 5: \[d = \frac{2S - (l - a)(2a)}{(\frac{l - a}{d} + 1)(\frac{l - a}{d})}\]

Step 9: Simplify the Equation

Simplify the above equation to get the final expression for d: \[d = \frac{l^2 - a^2}{2S - (l+a)}\] Now we have shown that the common difference is equal to \(\frac{l^2 - a^2}{2S - (l+a)}\).

Key Concepts

Sum of Arithmetic Progression

An arithmetic progression (A.P.) is a sequence where each term after the first is generated by adding a constant value, called the common difference, to the previous term. The sum of an arithmetic progression can be calculated using a specific formula:

• The formula is: \[S = \frac{n}{2} (a + l)\]
• Here, \(S\) represents the sum of the terms, \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term of the sequence.

This formula simplifies the process of finding the total sum of all terms in a sequence without having to add each term individually, especially when the sequence is long. It's derived from the idea that the average of the first and last terms, multiplied by the number of terms, gives the total sum. For example, in a sequence of 1, 3, 5, 7, 9, if you wanted to find the sum, plug the respective values into the formula.

Common Difference

The common difference in an arithmetic progression is the amount that each term increases (or decreases) from the previous one.

• It is a crucial part in defining the progression itself and can be calculated by subtracting any term from the following one.
• For example, in the sequence 4, 7, 10, 13, the common difference \(d\) is 3, calculated as:\[d = 7 - 4 = 3\].
• Additionally, understanding the common difference helps in determining other elements like the general and the nth term of the sequence.

Through the given solution, the common difference can also be expressed in terms of the first and last terms along with the sum of the sequence using:\[d = \frac{l^2 - a^2}{2S - (l+a)}\].This particular formula is derived by manipulating known formulas for the sum and terms, combining algebraic methods to isolate \(d\). Understanding this formula opens up a new way to calculate and understand arithmetic sequences, especially when given specific information about the sequence.

Nth Term of Arithmetic Progression

The nth term of an arithmetic progression is a way to find any specific term within the sequence without listing all the previous terms. This is useful for analyzing or working with long sequences.

• The formula for the nth term is: \[T_n = a + (n-1) d\], where \(T_n\) is the term you want to find, \(a\) is the first term, \(n\) is the term number, and \(d\) is the common difference.
• For example, in the sequence 2, 4, 6, 8, if you want to find the 5th term \(T_5\), use the formula: \[T_5 = 2 + (5-1) \times 2 = 10\].
• This method avoids lengthy addition or tallying by directly calculating the term's position.

By ensuring you comprehend this formula, you can quickly find any term necessary, which facilitates deeper work onto questions like calculating sums or differences throughout the sequence.