How do you find the first three terms of the arithmetic series #a_1=17#, #a_n=197#, and #S_n=2247#?

#1#st term is #a_1=17 , a_n=197# , common difference is #b#

And number of terms is #n#.

Mid term is # a_m=(17+197)/2=107 #

Sum # S_n+= a_m*n or 2247 = 107 * n :. n= 2247/107=21#

#a_n= a_1+(n-1)b or 197 = 17+ (21-1)b or 20b=180 # or

#b=180/20=9 :. a_2=a_1+b=17+9=26 #

# a_3=a_1+2b=17+2*9=17+18=35 #

First three terms are #17,26 and 35# [Ans]

An arithmetic sequence is the one in which the difference of the successive terms is the same

For example,

#1,3,5,7..# is an arithmetic sequence because the common difference is #2#. The symbol that i will use for the common difference here is #d#

Now lets solve the problem. For that we use the formula

#color(green)(S_n=n((a_1+a_n)/2)#

Where, #S_n# is the sum of the first #n# terms, #a_1# is the first term and #a_n# is the #n^(th)# term. We already knew the values, so apply them

#rarr2247=n((197+17)/2)#

#rarr2247=n(214)#

#color(green)(rArrn=21#

Now we know that the #21^(st)# term is #197#

To find the common difference, we use

#color(green)(a_n=a_1+(n-1)d#

#rarr197=17+(21-1)d#

#rarr180=20d#

#color(green)(rArrd=9#

Now we have come to an end to the answer. We add #9# to each of the successive terms and we get

#color(purple)(17,26,35....#

Hope this helps!!! ☺