Question

How do you find the answer to the sum of all the digits in all the numbers from 1 to 100

ex: 98 to 100 9+8+9+9+1+0+0=36

Answer 1

We can find the sum of all digits from 1 to 100 by

= (1 + 99) + (2 + 98) + (3 + 97) + (4 + 96) + (5 + 95) + (6 + 94) + (7 + 93) + (8 + 92) + (9 + 91) + (10 + 90) + (11 + 89) + (12 + 88) + (13 + 87) + (14 + 86) + (15 + 85) + (16 + 84) + (17 + 83) + (18 + 82) + (19 + 81) + (20 + 80) + (21 + 79) + (22 + 78) + (23 + 77) + (24 + 76) + (25 + 75) + (26 + 74) + (27 + 73) + (28 + 72) + (29 + 71) + (30 + 70) + (31 + 69) + (32 + 68) + (33 + 67) + (34 + 66) + (35 + 65) + (36 + 64) + (37 + 63) + (38 + 62) + (39 + 61) + (40 + 60) + (41 + 59) + (42 + 58) + (43 + 57) + (44 + 56) + (45 + 55) + (46 + 54) + (47 + 53) + (48 + 52) + (49 + 51) + (50 + 50)

= 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 100

= 100 × 50

= 5000

We can also find the sum of all the digits from 1 to 100 by :-

= 1/2 of 100²