(1)- Find the sum of 1+3+7... _ 57 (nth odd is (2n-1)
(2)- Find the sum of the first 28 counting numbers.
+128408 (1) 1 + 3 + 5 + ....+ 57
Note that the sum of the first 2 odd positive integers = 2^2
And the sum of the first 3 positive odds = 3^2
And
2n -1 = 57
2n = 58
n = 29
So 57 is the 29th positive odd
So.....the sum of 1 +3 + 5 + ....+ 57 is the sum of the first 29th odd numbers = 29^2 = 841
(2) Sum of 1st n counting numbers = n (n + 1) / 2
So....the sum of the first 28 counting numbers = 28 * 29 / 2 = 14 * 29 = 406
(1)- Find the sum of the first 28 odd couting numbers
(2)- Develop the formula of the sum of the first n counting numbers.
S= 1 + 2 + 3 + ... + (n-2) + (n-1) + n
(3)- Find the sum of the first 499 counting numbers
+128408 1. From above → 28^2 = 784
2. Sum of first n counting numbers
Note that we have
1 + 2 + 3 + 4 + .....+ n - 3 + n -2 + n -1 + n
Note that we can rearrange this as
(1+ n) + ( n -1 + 2) + ( n - 2 + 3) + ( n - 3 + 4) + ..... =
(n + 1) + ( n + 1) + (n + 1) + ( n + 1) + ....
We have n /2 of these pairs....so.....the sum is just n * ( n + 1) / 2
3. Sum of 1st 499 counting numbers = 499 * ( 500) /2 = 499 * 250 = 124,750
