Example Question - sequence terms
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Question on Sequence Terms from Binomial Expansion and Line Equations
<p>For the binomial expansion part, to find the 4th term from the end in the expansion of $\left( \frac{3}{2} - \frac{x^3}{6} \right)^7$, note that the 4th term from the end is equivalent to the 4th term from the beginning (or the $T_4$ term) because the binomial is symmetric.</p> <p>To find $T_4$, we use the binomial theorem which states $T_{k+1} = ^nC_k \cdot a^{n-k} \cdot b^{k}$.</p> <p>$T_4 = ^7C_3 \left( \frac{3}{2} \right)^{7-3} \left( -\frac{x^3}{6} \right)^3$</p> <p>$T_4 = 35 \cdot \left( \frac{3}{2} \right)^4 \cdot \left( -\frac{x^3}{6} \right)^3$</p> <p>$T_4 = 35 \cdot \frac{81}{16} \cdot -\frac{x^9}{216}$</p> <p>$T_4 = -\frac{35 \cdot 81 \cdot x^9}{16 \cdot 216}$</p> <p>$T_4 = -\frac{35 \cdot 81 \cdot x^9}{6^3 \cdot 6}$</p> <p>$T_4 = -\frac{35 \cdot 81 x^9}{6^4}$</p> <p>$T_4 = -\frac{945 \cdot x^9}{1296}$</p> <p>For the equation of lines passing through (1,2) and making an angle $30^\circ$ with the y-axis, the slope of the line is the tangent of the angle with the x-axis. Since the line makes a $30^\circ$ angle with the y-axis, it makes a $60^\circ$ angle with the x-axis. Hence, the slope $m$ is $\tan(60^\circ) = \sqrt{3}$.</p> <p>The equation of a line in slope-intercept form is $y = mx + b$.</p> <p>To find $b$, substitute $(1,2)$ (x, y) into the equation:</p> <p>$2 = \sqrt{3}(1) + b$</p> <p>$b = 2 - \sqrt{3}$</p> <p>Therefore, the equation of the line is:</p> <p>$y = \sqrt{3}x + (2 - \sqrt{3})$</p>
Calculation of Sequential Terms in a Recursive Sequence
<p>Nous allons calculer les trois termes suivants de la suite arithmétique définie par \( u_0 = 2 \) et \( u_{n+1} = 3u_n - 4 \) pour tout entier naturel \( n \).</p> <p>Le premier terme est déjà donné : \( u_0 = 2 \).</p> <p>Nous calculons le deuxième terme: \( u_1 = 3u_0 - 4 = 3 \cdot 2 - 4 = 6 - 4 = 2 \).</p> <p>Ensuite, le troisième terme: \( u_2 = 3u_1 - 4 = 3 \cdot 2 - 4 = 6 - 4 = 2 \).</p> <p>Enfin, le quatrième terme: \( u_3 = 3u_2 - 4 = 3 \cdot 2 - 4 = 6 - 4 = 2 \).</p> <p>Les trois termes suivant le premier sont donc tous égaux à 2.</p>
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