Solution:Series I: a, 156, 110, 72, 42, 20, b
Find differences of known terms:
156 − 110 = 46
110 − 72 = 38
72 − 42 = 30
42 − 20 = 22
Differences: 46, 38, 30, 22
Observe pattern:
46 − 38 = 8
38 − 30 = 8
30 − 22 = 8
So differences decrease by 8 each time.
Therefore previous difference: 46 + 8 = 54
So:
a − 156 = 54
a = 210
Now continue pattern after 22 : (22 − 8 = 14)
So:
20 − b = 14
b = 6
Thus:
a = 210
b = 6
Series II:
a + c
a
a − c
a − 2c
a − 3c
b
d
Substitute a = 210 and b = 6:
210 + c
210
210 − c
210 − 2c
210 − 3c
6
d
This is clearly an Arithmetic Progression with common difference = −c
So: 210 − 3c = 6
Solve:
210 − 3c = 6
3c = 204
c = 68
Next term:
d = b − c
d = 6 − 68
d = −62
Finding (b − d) = 6 − (−62) = 6 + 62 = 68