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#

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

#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)#