Question

How do you find `S_n` for the arithmetic series given `a_n=-53/12, d=-1/4, n=20`?

Answer 1

`-43 5/6" " ->" " 263/6`

Explanation:

`color(blue)("Consider the structure of an Arithmetic sequence")`

Let any one term count by `i`
Let any one term be `a_i`
Let the total count be `n`
Let the difference between terms be `d`
Let the sum of the series be `S_n`

Set `d=-1/4" "` then we have:

`a_i->a_1`
`a_i->a_2=a_1-1/4`
`a_i->a_3=a_1-1/4-1/4`
`a_i->a_4=a_1-1/4-1/4-1/4`
:
:

This implies that for any `a_i`

`a_i=a_1-1/4(i-1)`

Thus `color(brown)(a_n=a_1-1/4(n-1))color(green)(" "->" "a_20=-53/12=a_1-1/4(20-1))`

`=>a_1=-53/12+1/4(19)`

`=>a_1=1/3`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Check: `a_20 = 1/3-1/4(19) = -53/12`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The sum `S_n=" total count "xx"mean value"`

Mean value `->("first term + last term")/2`

`S_20=(20(1/3-53/12))/2 = -43 5/6 -> 263/6`