Answer:
[tex]S_{10}[/tex] = - 30
Step-by-step explanation:
For a given arithmetic sequence the n th term formula is
[tex]t_{n}[/tex] = [tex]t_{1}[/tex] + (n - 1)d
where d is the common difference and [tex]t_{1}[/tex] the first term
We have to find d and [tex]t_{1}[/tex]
from the given information we can write 2 equations and solve for d and [tex]t_{1}[/tex]
[tex]t_{6}[/tex] = [tex]t_{1}[/tex] + 5d = - 4 → (1)
[tex]t_{10}[/tex] = [tex]t_{1}[/tex] + 9d = - 12 → (2)
subtract (1) from (2) term by term
4d = - 8 ⇒ d = - 2
substitute d = - 2 in (1)
[tex]t_{1}[/tex] - 10 = - 4 ⇒ [tex]t_{1}[/tex] = - 4 + 10 = 6
The sum to n terms of an arithmetic sequence is
[tex]S_{n}[/tex] = [tex]\frac{n}{2}[/tex][2[tex]t_{1}[/tex] + (n - 1)d ], hence
[tex]S_{10}[/tex] = 5[(2 × 6) + (9 × - 2) ] = 5(12 - 18) = 5 × - 6 = - 30
