First we will list the data we are given.
The sum of the first 8 terms of the A.P is = #100#.
The sum of the first 19 terms is = #551#.
Now Let The First term of the A.P be #a#, and the common difference be #d#.
So, According to the problem,
#color(white)(xxx)8/2[a + a + (8 - 1) d] = 100#
#rArr 4[2a + 7d] = 100#
#rArr 2a + 7d = 25#........................................(i)
And, #19/2[a + a + (19 - 1)d] = 551#
#rArr 19/2 [2a + 18d] = 551#
#rArr 19[a + 9d] = 551#
#rArr a + 9d = 29#..........................(ii)
Now, Just Solve For #a# and #d#.
FIrst, Multiply eq(ii) with #2#.
So, we get,
#2a + 18d = 58#............................(iii)
Now. Subtract eq(i) from eq(iii).
So, We get,
#color(white)(xxx)cancel(2a) + 18d cancel(- 2a) - 7d = 58 - 25#
#rArr 11d = 33#
#rArr d = 3#
Now, Substitute #d = 3# in eq(i).
So, We get,
#color(white)(xxx)2a + 7 * 3 = 25#
#rArr 2a + 21 = 25#
#rArr 2a = 25 - 21#
#rArr 2a = 4#
#rArr a = 2#
So, Now we can form the A.P.
The AP will be #a, a + d, a + 2d, a + 3d,............................,a + (n - 1)d.#
So, The Finalised A.P. is :-
#2, 5, 8, 11, 14, ............................................#
