Respuesta :
Answer:
The value of a, d, and U9 is "[tex]\bold{ \frac{27}{2}, - \frac{7}{2}, \ and , - \frac{29}{2}}[/tex]".
Explanation:
Given:
[tex]U_5=-\frac{1}{2}\\\\S_7=21[/tex]
Find:
a, d, U9=?
formula:
[tex]T_n=a+(n-1)d\\\\S_n=\frac{n}{2}(2a+(n+1)d)[/tex]
Solve:
[tex]\to U_5 = a+4d= -\frac{1}{2}\\\ \to 2a+8d= -1\\\ \to 2a= -1-8d....(a)\\\\ \to S_7 =\frac{7}{2}(2a+(7-1)d)\\\\ \to S_7 =\frac{7}{2}(2a+6d)\\\\ \to 21= \frac{7}{2} (2a+6d)\\\\ \ put \ the \ 2a \ value\\\\ \to 21= \frac{7}{2} ((-1-8d)+6d)\\\\ \to 21= \frac{7}{2} (-1-8d+6d)\\\\ \to 21= \frac{7}{2} (-1-2d)\\\\ \to \frac{21 \times 2}{7}= -1 -2d\\\\ \to 6 = -1 -2d\\\\ \to 2d=-7\\\\ \to d= -\frac{7}{2}[/tex]
put the value of d in equation a:
[tex]\to 2a= -1-8 \times \frac{-7}{2}\\\\\to 2a= -1+ 28\\\\\to 2a= 27\\\\\to a =\frac{27}{2}\\[/tex]
[tex]\to \bold{U_9 = a+8d}\\\\[/tex]
put a and b value in above equation:
[tex]\to U_9 = \frac{27}{2}+8\times\frac{-7}{2} \\\\\to U_9 = \frac{27}{2}- 28\\\\\to U_9 = \frac{27-56}{2}\\\\\to U_9 = \frac{-29}{2}\\\\[/tex]
