Inactive Tutor answered • 03/03/22
1+5+9+13+17+ 21+25 +29 +33+ 37+ 41+ 45 + 49 +53 +57+61 +65 + 69 + 73 + 77 +81 + 85..+89.+ 93 + 97
= 1225
97+1 = 98
93+5 = 98
89+9 =98
pair up 1st and last
2nd and 2nd to last
3rd and 3rd to last
etc.
45+53 =98
they all sum to 98
there are 12 pairs summing to 98.
12x98= 1176
but add in the middle number that didn't have a pair, 49 to get 1225
there's 25 numbers, the 13th integer is 49 = the middle number
Gauss' formula is sum = n(n+1)/2 = 98(97)/2 = 49(97) = 4553 = sum of 1+2+...+ 96 + 97
which is very roughly 4 times sum of every 4th integer
but another precise method is subtract 1, divide by 4, apply Gauss' formula, then multiply by 4 and add back1
subtract 1 from each integer in the given sum to get 0,4,8,12,16,.....96
now divide by 4 to get 1,2,3, ...24
apply Gauss' formula 24(25)/2 = 12(25) =300
multiply 300 by 4 to get 1200
add back the 1 to each of the 25 integers. 1200+25 = 1225
