The formula of a sum of terms of an arithmetic sequence:
[tex]S_n=\dfrac{t_1+t_n}{2}\cdot n[/tex]
We have
[tex]S_{12}=21,\ t_{12}=10,\ n=12[/tex]
Substitute:
[tex]21=\dfrac{t_1+10}{2}\cdot12[/tex]
[tex]21=(t_1+10)\cdot6[/tex] use the distributive property: a(b + c) = ab + ac
[tex]21=6t_1+60[/tex] subtract 60 from both sides
[tex]-39=6t_1[/tex] divide both sides by 6
[tex]t_1=-\dfrac{39}{6}\\\\t_1=-\dfrac{13}{2}[/tex]
The explicit form of an arithmetic sequence:
[tex]t_n=t_1+(n-1)d[/tex]
d - common difference
[tex]d=t_{n+1}-t_n\to 11d=t_{12}-t_1[/tex]
Substitute:
[tex]11d=10-\left(-\dfrac{13}{2}\right)[/tex]
[tex]11d=\dfrac{20}{2}+\dfrac{13}{2}[/tex]
[tex]11d=\dfrac{33}{2}[/tex] divide both sides by 11
[tex]d=\dfrac{3}{2}[/tex]
[tex]t_2=t_1+d\to t_2=-\dfrac{13}{2}+\dfrac{3}{2}=-\dfrac{10}{2}=-5[/tex]
Answer:
[tex]t_1=-\dfrac{13}{2}\ and\ t_2=-5[/tex]
