Question

Is this correct? The writing in red is all my work. It's about sigma notation.

Answer 1

`\boxed{\displaystyle \sum_(k=1)^(10)(21.5 - 1.5n)}`

Explanation:

If you have an arithmetic sequence

a₁ + a₂ + a₃ + … + aₙ

the general sigma notation for the sum of the first n terms is

`\displaystyle \sum_(k=1)^(n) a_(k)\\k \text{ is the index or counter}\\n \text{ is the number of the last term}\\a_(k) \text{ is the general formula for each term}`

k = 1 means that you start at the first term and keep incrementing until k = n.

The formula for the nth term of an arithmetic sequence is

aₙ = a₁ + (n - 1)d

In your sequence,

a₁ = 20 and d= -1.5, so

aₙ = 20 - 1.5(n - 1) =20 - 1.5n + 1.5 = 21.5 - 1.5n

Thus, the sigma notation for your sequence is

`\boxed{\displaystyle \sum_(k=1)^(10)(21.5 - 1.5n)}`

Answer 2

Explanation:

OK. It's an arithmetic sequence:

`a_1=20,\ a_2=18.5,\ a_3=17,\ ...\\\\a_1=20,\ d=-1.5`

The explicit formula of an arithmetic sequence:

`a_n=a_1+(n-1)d`

Substitute:

`a_n=20+(n-1)(-1.5)=20-1.5n+1.5=21.5-1.5n`

The sigma notation of the sum of the first ten terms:

`\sum\limits_(n=1)^(10)(21.5-1.5n)`

What are your mistakes:

`\sum\limits_(n=20)^(6.5)\to\boxed{n=20},\ \boxed{6.5}`

The first ten terms, not from 20th to 29th (you wrote 6.5?)

`\sum\limits_(n=1)^(10) - \text{the sum for n = 1 to n = 10}`