find the sum of the series?

I think that this approach is most straigtforward and easiest to remember:

there are (725-19) 706 numbers in the series.

Why minus 19 instead of minus 20? look at the series from 20+21. There are two elements in the series. Frirt minus last takes away one too many to get the count of the number of elements.

average number in the series = (20+725)/2

sum of the series = (745)(706)/2

next series we are stepping up by twos...

number of elements in the series.

(801-11)/2 + 1

average value

(801+11)/2

sum of the series.

((790)/2 + 1)(812)/2

As they gave you these formula...

S = n(n+1)/2 This is the formula from 1 to n so S(725) - S(19) should give the same answer as above...

725*726 / 2 - 19*20/2 = (725-19)(725 + 20) /2

(725*726 - 19*20)/2 = (725^2 + 725 -19*20) /2

(725*726 - 19*20)/2 = (725 * 726 -19*20) /2

and

S= n(n+1) .... S(401) - S(5)

401^2 - 5^2 = ((790)/2 + 1)(812)/2

S = n(n+1)/2 is the sum of the arithmetic series 1 +2 +3 +4 +.......+n

20 + 21 +22+..........+725 =1 +2 +3+..........+725 - (1` + 2 +3 + ........+19)

=725*726/2 - 19*20/2 = 249571 - 190 = 249381.

The second one is similar, S = n^2 is the sum of 1 +3 +5 +...........+(2 n - 1 )

So 11 + 13 +.........801 = (1 + 3 + .........+801) - (1 + 3 + 5 + 7 + 9)

If n is the number of terms in the first series then 2n - 1 = 801 so that n = 401, and clearly the number of terms in the second series is 5. Our answer is therefore 401^2 - 5^2.