Sequences & Series Question - None

Question

Let $\alpha$, $\beta$ and $\gamma$ be the roots of the equation $x^3 - 4x + 1 = 0$. If $T_n = \alpha^n + \beta^n + \gamma^n$, $n \geq 1$, then which of the following is(are) true?
[Note: $[k]$ denotes greatest integer function less than or equal to $k$.]

Accepted Answer

Step 1: The given equation is $x^3 - 4x + 1 = 0$, and we are asked to find the values of $T_n = \alpha^n + \beta^n + \gamma^n$, where $\alpha$, $\beta$, and $\gamma$ are the roots of the equation.

Step 2: We can use Newton's Sums to find the values of $T_n$. Newton's Sums state that for a polynomial $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0$, the sum of the $n$th powers of the roots is given by $T_n = e_1T_{n-1} - e_2T_{n-2} + \ldots + (-1)^{n+1}ne_n$, where $e_i$ is the $i$th elementary symmetric polynomial in the roots.

Step 3: For the given equation $x^3 - 4x + 1 = 0$, we have $e_1 = 0$, $e_2 = -4$, and $e_3 = -1$. Using Newton's Sums, we can find $T_1$, $T_2$, and $T_3$. We have $T_1 = 0$, $T_2 = -4$, and $T_3 = 0 + 4(0) - 3(1) = -3$.

Step 4: We can now use the recurrence relation $T_n = e_1T_{n-1} - e_2T_{n-2} + e_3T_{n-3}$ to find $T_4$, $T_5$, and $T_6$. We have $T_4 = 0T_3 - (-4)T_2 + (-1)T_1 = 16$, $T_5 = 0T_4 - (-4)T_3 + (-1)T_2 = 4$, and $T_6 = 0T_5 - (-4)T_4 + (-1)T_3 = -4(16) + 3 = -61$.

Step 5: We can now check the given options. For option (a), we have $T_6 - 4T_4 = -61 - 4(16) = -93 \neq 3$, so option (a) is false. For option (b), we need to find $T_{96}$, $T_{99}$, and $T_{102}$. We can use the recurrence relation to find these values.

Step 6: After calculating $T_{96}$, $T_{99}$, and $T_{102}$, we find that $T_{96} + 2T_{99} + T_{102} = 16T_{98}$ is false, and $T_{96} + 2T_{99} + T_{102} = 16T_{100}$ is true. For option (d), we have $[\alpha] + [\beta] + [\gamma] = -2$, which is false since the sum of the roots is 0.

Answer: $T_{96} + 2T_{99} + T_{102} = 16T_{100}$ is true.

Key Concept: Newton's Sums, recurrence relations, and properties of polynomial roots.

Question:

Step-by-Step Solution

Master Sequences & Series with Mathbee