Respuesta :
Answer:
(a) The value of [tex]s_z[/tex] is (z+1)(3-z).
(b) The next term in the sequence is -2.
Step-by-step explanation:
(a)
It is given that arithmetic sequence that starts with an initial index of 0.
The initial term is 3 and the common difference is -2.
[tex]a_0=3[/tex]
[tex]d=-2[/tex]
We need to find the value of [tex]s_z[/tex].
[tex]s_z=\sum_{n=0}^{n=z}(a+nd)[/tex]
where, a is initial term and d is common difference.
[tex]s_z=\sum_{n=0}^{n=z}(3-2n)[/tex]
The sum of an arithmetic sequence with initial index 0 is
[tex]s_n=\frac{n+1}{2}[2a+nd][/tex]
where, a is initial term and d is common difference.
Substitute n=z, a=3 and d=-2 in the above formula.
[tex]s_z=\frac{z+1}{2}[2(3)+z(-2)][/tex]
[tex]s_z=\frac{z+1}{2}[2(3-z)][/tex]
[tex]s_z=(z+1)(3-z)[/tex]
Therefore the value of [tex]s_z[/tex] is (z+1)(3-z).
(b)
The given arithmetic sequence is
7, 4, 1, ...
We need to find the term in the sequence.
In the given arithmetic sequence the first term is
[tex]a=7[/tex]
The common difference of the sequence is
[tex]d=a_2-a_1\Rightarrow 4-7=-3[/tex]
The first term is 7 and common difference is -3.
Add common difference in last given term, i.e., 1, to find the next term of the sequence.
[tex]1+(-3)=1-3=-2[/tex]
Therefore the next term in the sequence is -2.
