What is the sum of the arithmetic sequence 9,14,19... if there are 34 terms?

Let the sequence position be `i`
Let the sequence position of the last term be `n`

Let the difference between each term be `d`

Let the ith term in the sequence be `a_i`
Let the last term in the sequence be `a_n`

So
`a_i->a_1=9`
`a_i->a_2=14`
`a_i->a_3=19`
`a_i->a_n=a_34=?" "larr "the last term in the sequence"`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)("Investigating the 'common difference' between each term")`

`14-9= 5`
`19-14=5`

`color(green)("the difference between each successive term is "d=5)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Determine the value of the 34th term")`

So if `a_i->a_1 = 9" "` then `" "a_i->a_0 = 9-5=4`

So
`a_1=a_0+d`
`a_2=a_0+d+d" "->" "a_0+2d`
`a_3=a_0+d+d+d" "->" "a_0+3d`

so `a_i=a_0+(ixxd)`

and `a_n=a_0+(nxxd)`

So `a_n=a_34=a_0+(34xxd)" "->" "4+(34xx5) = 174`
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`color(blue)("Determine the sum of the 34 terms")`

Consider again the:
`a_1=a_0+d`
`a_2=a_0+d+d" "->" "a_0+2d`
`a_3=a_0+d+d+d" "->" "a_0+3d`

The sum of `a_1 to a_3` is:

`a_1->a_0+d`
`a_2->a_0+2d`
`a_3->underline(a_0+3d)" "larr" add"`
`" "3a_0+6d`

Which is the same as:

`3a_0+(d+2d+3d)`

`3a_0+d(1+2+3)`

`3a_0+d(3xx"Mean value")`

`(ixxa_0) + d(ixx("first count+last count")/2)`

`(ixxa_0)+d(ixx(1+i)/2)`
'.....................................................
So for a count of `n` terms we have

`color(green)((na_0)+d(nxx(1+n)/2))`

`color(red)("Notice that in the end I did not need the value of the last term")`
'.................................................
`n=34`
`d=5`
`a_0=4`

sum of the sequence is `(34xx4)+5(34xx(1+34)/2)`

`color(green)(=136+2975=3111)`