Exercice 9 --- (id : 588)
Suites: Exercice 9
correction
1 $U_9=U_4+(9-4)r$ $\iff r=\dfrac{U_9-U_4}{5}=\dfrac{28-13}{5}=3$
$U_4=U_0+4r$ $\iff U_0=U_4-4r=13-12=1$
2 $U_n=61 \iff U_0+nr=61$ $\iff 1+3n=61$ $\iff 3n=60$ $\iff n=\dfrac{60}{3}=20$
3
$S=U_0+U_1+...+U_{n-1}$
a $S=\dfrac{n-1-0+1}{2}(U_0+U_{n-1})$ $\iff S=\dfrac{n}{2}(1+1+3(n-1))$ $\iff S=\dfrac{n(3n-1)}{2}=\dfrac{3n^2-n}{2}$
b $$\begin{align*} &1+4+7+...+58+61 \\ &=(1+3\times 0)+(1+3\times1)+(1+3\times2)+...+(1+3\times19)+(1+3\times20) \\ &=\sum\limits_{k=0}^{20}{(1+3k)}=\sum\limits_{k=0}^{20}{U_k} \\ &=\dfrac{20-0+1}{2}(U_0+U_{20}) \\ &=\dfrac{21}{2}(1+61)= 21\times31=651 \end{align*}$$