\large\displaystyle\sum\limits_(k=1)^(n){{a_k}}The series 25 + 75 + 125 + ... + 775 + 825 can be written using sigma notation.Write an expression for ak in terms of k. And find …

Question

`\large\displaystyle\sum\limits_(k=1)^(n){{a_k}}`The series 25 + 75 + 125 + ... + 775 + 825 can be written using sigma notation.Write an expression for ak in terms of k. And find n

Answer 1

`\large \boxed{\sf \ \ \sum_(k=1)^(17) (-25+50k)=25\cdot 17^2=7225 \ \ }`

Explanation:

Hello,

75 - 25 = 50

125 - 75 = 50

so

`a_0=-25`

`a_1=-25+50=25`

`a_n=-25+50n`

Then we can write

`\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 (n(n+1))/(2)\\\\=(-50\cdot n+50\cdot n(n+1))/(2)\\\\=(50)/(2)(-n+n(n+1))\\\\=25(-n+n^2+n))\\\\=25n^2`

to note that

`a_(17)=-25+50*17=-25+850=825`

because

-25 + 50n = 825

<=> 50n = 850

<=> n = 17

Hope this helps.

Do not hesitate if you need further explanation.

Thank you