Sum of n terms , Sn

= n/2 (2a + (n-1)d)

where n is the number of terms , a is the first term and d is the common difference.


Sum of first twenty terms, S20 = 50

20/2 (2a + (20 - 1)d) = 50

10(2a + 19d) = 50

2a + 19d = 5

2a = 5 - 19d ①


Sum of next twenty terms

= Sum of first forty - sum of first twenty

= S40 - S20

= -50

So,

40/2 (2a + (40 - 1)d) - 50 = -50

20(2a + 39d) = 0

2a + 39d = 0 ②


Sub ① into ②,


5 - 19d + 39d = 0

20d = -5

d = -¼


Then, 2a = 5 - 19(-¼)

2a = 39/4

a = 39/8 (or 4⅞)

Therefore, sum of first 100 terms, S100

= 100/2 (2a + (100 - 1)d)

= 50 (2a + 99d)

= 50(39/4 + 99(-¼))

= 50(-15)

= -750