ANSWER :
First sum : -14,625
Second sum : 11,390
EXPLANATION :
The sum formula of an arithmetic series is :
[tex]S_n=\frac{n}{2}(a_1+a_n)[/tex]
where Sn = sum
n = number of terms
a1 = first term
an = last term
From the problem, we have the series :
[tex]5+0+(-5)+...+(-380)[/tex]
We have the first term, a1 = 5 and the last term, an = -380.
But we don't know the number of terms.
Using the nth term formula of an arithmetic series.
[tex]a_n=a_1+d(n-1)[/tex]
The difference in the series is 0 - 5 = -5
Let's solve for the value of n :
[tex]\begin{gathered} a_{n}=a_{1}+d(n-1) \\ -380=5-5(n-1) \\ -380-5=-5(n-1) \\ -385=-5(n-1) \\ \frac{-385}{-5}=n-1 \\ 77=n-1 \\ 77+1=n \\ n=78 \end{gathered}[/tex]
So there are 78 terms.
Now use the sum formula :
[tex]\begin{gathered} S_n=\frac{78}{2}(5-380) \\ S_n=-14625 \end{gathered}[/tex]
The sum is -14,625
For the second sum, we have :
[tex]\sum_{n\mathop{=}1}^{85}(3j+5)[/tex]
The first term will be :
[tex]3(1)+5=8[/tex]
Solve for the last term at j = 85
[tex]3(85)+5=260[/tex]
To summarized, we have :
a1 = 8
an = 260
n = 85
Using the sum formula above :
[tex]\begin{gathered} S_n=\frac{85}{2}(8+260) \\ S_n=11390 \end{gathered}[/tex]
The sum is 11,390
