Question

If \(\mathrm{a}_{\mathrm{r}}\) is the coefficient of \(\mathrm{x}^{\prime}\) in the expression of \(\left(1+x+x^{2}\right)^{n}\), then the value of \(a_{1}+4 a_{4}+7 a_{7}+10 a_{10}+\ldots \ldots . .\) is equal to (1) \(3^{n-1}\) (2) \(n 3^{n-1}\) (3) \(2^{n}\) (4) \(\frac{2^{n}}{3}\)

Short answer

The value is \( n \cdot \ 3^{n-1} \).

Step-by-step solution

Understand the Problem

The problem involves the expansion of \( \left(1 + x + x^2 \right)^n \) and finding the sum of the coefficients of specific terms in this expansion.

General Form of the Expansion

The expression \( \left(1 + x + x^2 \right)^n \) can be expanded using the binomial theorem. Each term in the expansion has the form \( a_r x^r \), where \( a_r \) is the coefficient of \( x^r \).

Focus on Specific Coefficients

We need to find the sum of the coefficients \( a_1 + 4a_4 + 7a_7 + 10a_{10} + \ldots \). This means identifying the coefficients of terms where the exponent is of the form of multiples of 3 plus 1 (since the sequence progresses as \( a_1 + a_{1+3} + a_{1+6} + \ldots \)).

Simplify the Series

To simplify, observe that \( 1 + x + x^2 \) can be considered as roots of unity (i.e., summing to one when raised to some powers). Instead of lengthy expansion, consider each term being accumulated in the sum.

Sum of Coefficients Analysis

We know that \( 1 + x + x^2 = 0 \) will repeat periodically for every \( n \). Hence, the form follows periodic structure and pattern; applying binomial sum techniques.

Final Calculation

For the sequence chosen, the arithmetic sum can be generalized in compact form: Each subsequent term forms an arithmetic progression in multiplicative forms. Overall, \( the required sum = n * 3^{n-1} \)

Key concepts

Binomial Expansion

The foundation of solving this problem lies in understanding the binomial expansion. In the context of our problem, the given expression is \((1 + x + x^2)^n\). Binomial expansion deals with expanding expressions that are raised to a power, for example, \((a + b)^n\). Using the binomial theorem, we can express this as a sum of terms involving binomial coefficients: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]In our problem, we have three terms inside the parenthesis instead of two. Therefore, the expansion will include terms involving all powers of \(1, x,\) and \(x^2\). Each of these terms will contribute to the coefficients of \(x^r\) in the expansion. Understanding this expansion helps in identifying the individual coefficients that are being summed up in the final result. The general form of a term in this expansion is \[ a_r = \sum_{i+j+k=n, \ i,j,k \geq 0} \binom{n}{i,j,k} x^{i+2j+k} \]where \(\binom{n}{i,j,k}\) represents the trinomial coefficient.

Series Summation

After understanding the forms of coefficients in our expanded expression, the next important concept is summing up specific coefficients to find the result. In the given problem, we are asked for the sum \(a_1 + 4a_4 + 7a_7 + 10a_{10} + \ldots\)The sequence we observe is of the form \((a_1 + a_4 + a_7 + \ldots)\). Notice the arithmetic pattern in the series \( 1, 4, 7, 10, \ldots\) which can be formed using the arithmetic sequence formula \[ a_r = 1 + (r - 1) * 3 \]where \(r\) represents the progression in the summation sequence. Here, we are essentially looking at coefficients of terms where the exponent is a multiple of 3, plus 1. Summing these coefficients is key to reaching the answer. By representing the sum in an arithmetic sum form, periodic structures in polynomial expansions simplify further calculations. Using such forms in series summation helps to manage complex polynomial terms by identifying patterns and using them for easy calculations.

Coefficients of Polynomial Terms

To solve the problem, an understanding of how coefficients of polynomial terms behave in expansions is crucial. Within an expression like \((1 + x + x^2)^n\), coefficients represent the numbers in front of each term when fully expanded. Given our target, we must identify terms with specific exponents. Each coefficient we are summing (i.e., \(a_1, a_4\), etc.) first needs identification in the expansion. This identification follows the comprehensive step-by-step polynomial expansions, aiming towards efficiently characterizing all possible terms. Analyzing coefficients come down to breaking each term's value to simpler binomial sum steps. By leveraging repeated patterns (roots of unity properties), we repeatedly use binomial summation forming periodic functions. In conclusion, the coefficient summation logic for our problem simplifies as follows: \((n[3^{n-1}])\) hence gives the correct answer being option \((2).\)