Problem 17 - Algebra for JEE ADVANCED

Question

If the sum of \(n\) terms of an A.P. is given by \(S_{n}=a+b n+c n^{2}\), where \(a, b, c\) are independent of \(n\), then a. \(a=0\) b. common difference of A.P. must be \(2 b\) c. common difference of A.P. must be \(2 c\) d. first term of A.P. is \(b+c\)

Step-by-Step Solution

Answer

a. \(a = 0\); c. Common difference is \(2c\); d. First term is \(b+c\).

Step 1: Write Sum of first n terms formula

In an arithmetic progression (A.P.), the sum of the first \(n\) terms, \(S_n\), can be expressed as \(S_n = \frac{n}{2} (2a_1 + (n-1)d)\), where \(a_1\) is the first term and \(d\) is the common difference.

Step 2: Compare to given expression

We compare the general sum formula \(S_n = \frac{n}{2} (2a_1 + (n-1)d)\) to the given expression \(S_n = a + bn + cn^2\). This comparison gives insight into what's the first term and the common difference of the A.P.

Step 3: Set up equations from coefficients

Since both expressions must be equivalent, equate the coefficients of similar terms:- Constant term (equating coefficients of 1): There is no constant term in \(\frac{n}{2} (2a_1 + (n-1)d)\), thus \(a = 0\).- Linear term (equating coefficients of \(n\)): The coefficient of \(n\) is \(b = \frac{d}{2}\).- Quadratic term (equating coefficients of \(n^2\)): The coefficient of \(n^2\) is \(c = \frac{d}{2}\).

Step 4: Solve for first term and common difference

Using the relationships from Step 3, - Since \(c = \frac{d}{2}\), we find that \(d = 2c\).- The first term \(a_1 = \text{coefficient of } n\text{'s linear part} = b + c\), combining the parts of the sum formula and the polynomial structure.

Key Concepts

Sum of n terms

To understand the sum of the first \(n\) terms of an arithmetic progression (A.P.), it's essential to grasp the formula and its components. Here, we're given a specialized formula: \(S_n = a + bn + cn^2\) where \(a\), \(b\), and \(c\) are constants, representing different coefficients for the numbers in the sequence. This format might look a bit different from what we're used to, but its essence mirrors the standard sum formula for A.P.:

• Standard formula: \(S_n = \frac{n}{2} (2a_1 + (n-1)d)\)

The completed polynomial format

• provides a sum formula that includes a quadratic term (\(cn^2\)), not commonly seen.


• shows that \(a = 0\), because generally, only linear and quadratic terms determine changes in sum.

By comparing these two equations, we can derive important characteristics about the progression's structure, like understanding how each term contributes to the overall sum. By checking coefficients of each term, such as \(b\) and \(c\), we see how they affect our arithmetic sequence.

Common difference

The common difference \(d\) is a critical feature of an arithmetic progression. It's the consistent interval, or space, between each term in the sequence, and can be determined through the sum formula.Compare the coefficients in the given sum format \(S_n = a + bn + cn^2\) to those in the typical one: \(S_n = \frac{n}{2} (2a_1 + (n-1)d)\). Upon equating, it's clear that:

• The coefficient of the linear term \(n\) is \(b\), which links to half the common difference, so \(b = \frac{d}{2}\).


• The coefficient of the quadratic term \(n^2\) is \(c\), which also connects with \(d\) as \(c = \frac{d}{2}\).

Thus, solving these, we find \(d = 2c\). Therefore, the common difference is twice the value of \(c\). This explains not only the interval between terms but also hints at how stretched or compressed our A.P. is.

First term

The journey to find the first term of an arithmetic progression from a given sum expression involves a few algebraic steps. Once we know the common difference, identifying the first term \(a_1\) becomes straightforward. By definition, the first term \(a_1\) is derived from the linear part of the sum expression, following our comparison technique with the known formula:

• We know that from \(S_n = a + bn + cn^2\), the linear and quadratic terms are linked with the A.P.'s features.


• The coefficient \(b + c\) gives us the actual first term, aligning with the standard sum formula.

So \(a_1 = b + c\), thus showing that the combination of linear influences (\(b\)) and quadratic connections (\(c\)) defines the initial value of our sequence. This highlights how the sequence starts based on the initial terms' alignment with the sum features.