Correction Exercice 11 Page 18 - Maîtriser- Math Tome 2 - 2éme

 Correction Activité 11 Page 18

a) Dans une suite arithmétique de raison $r$, le $n$-ème terme $u_n$ est donné par $u_n=u_1+(n-1)r$. Dans ce cas, on a la somme des 20 premiers termes $S_{20}$ qui est égale à 610. Ainsi,

$${�}_{20}=\frac{({�}_1+{�}_{20})\times 20}{2}=10({�}_1+{�}_{20})=610.$$

Comme la raison de la suite est 3, alors $u_{20} = u_1+19\times 3 = u_1+57$. Donc,

$${\mathrm{<span\; style="background-color:\; white;">�</span>}}_{\mathrm{<span\; style="background-color:\; white;">20</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">10</span>}<span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">�</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">+</span><span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">�</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">57</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">20</span>}{\mathrm{<span\; style="background-color:\; white;">�</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">570</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">610</span>}<span\; style="background-color:\; white;">,</span>$$

ce qui donne $u_1 = 2$.

b) La somme des 100 premiers termes $S_{100}$ est donnée par

$${\mathrm{<span\; style="background-color:\; white;">�</span>}}_{\mathrm{<span\; style="background-color:\; white;">100</span>}}<span\; style="background-color:\; white;">=</span>\frac{<span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">�</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">+</span>{\mathrm{<span\; style="background-color:\; white;">�</span>}}_{\mathrm{<span\; style="background-color:\; white;">100</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">100</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">50</span>}<span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">�</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">+</span>{\mathrm{<span\; style="background-color:\; white;">�</span>}}_{\mathrm{<span\; style="background-color:\; white;">100</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">50</span>}<span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">�</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">+</span>{\mathrm{<span\; style="background-color:\; white;">�</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">99</span>}\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">50</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">99</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">5050.</span>}$$

Ainsi, la somme des 100 premiers termes de cette suite est 5050.