Answer:
[tex]\large \boxed{\sf \ \ \sum_{k=1}^{17} (-25+50k)=25\cdot 17^2=7225 \ \ }[/tex]
Step-by-step explanation:
Hello,
75 - 25 = 50
125 - 75 = 50
so
[tex]a_0=-25[/tex]
[tex]a_1=-25+50=25[/tex]
[tex]a_n=-25+50n[/tex]
Then we can write
[tex]\displaystyle \sum_{k=1}^n a_k=\sum_{k=1}^n (-25+50k)\\\\=-25\codt \sum_{k=1}^n 1 + 50\cdot \sum_{k=1}^n k\\\\=-25\cdot n+50\cdot \dfrac{n(n+1)}{2}\\\\=\dfrac{-50\cdot n+50\cdot n(n+1)}{2}\\\\=\dfrac{50}{2}(-n+n(n+1))\\\\=25(-n+n^2+n))\\\\=25n^2[/tex]
to note that
[tex]a_{17}=-25+50*17=-25+850=825[/tex]
because
-25 + 50n = 825
<=> 50n = 850
<=> n = 17
Hope this helps.
Do not hesitate if you need further explanation.
Thank you
