For the sequence 1 + 2 + 3 + ... + 48 + 49 + 50, pair up the lowest and highest terms, then the next lowest/highest, etc. Each of those pairs add up to the same value: (1 + 50) + (2 + 49) + (3 + 48) + ... <-- 25 pairs = 51 + 51 + 51 + ... <-- 25 times = 51 * 25

there are in all 50 numbers from 4 to 50 ( since 4x 50 = 200 ) , the arithmetic average of the number series is : ( 4 + 200 ) / 2 = 204 / 2 = 102 . so the sum of all 50 numbers : 50 numbers x average value = 50 x 102 = 5100 .

## Answers

4 + 8 + 12 + ... + 200

= 4(1 + 2 + 3 + ... + 50)

For the sequence 1 + 2 + 3 + ... + 48 + 49 + 50, pair up the lowest and highest terms, then the next lowest/highest, etc. Each of those pairs add up to the same value:

(1 + 50) + (2 + 49) + (3 + 48) + ... <-- 25 pairs

= 51 + 51 + 51 + ... <-- 25 times

= 51 * 25

So 4 + 8 + 12 + ... + 200

= 4(51 * 25)

= 4 * 25 * 51

= 100 * 51

= 5100

there are in all 50 numbers from 4 to 50 ( since 4x 50 = 200 ) , the arithmetic average of the number series is :

( 4 + 200 ) / 2 = 204 / 2 = 102 .

so the sum of all 50 numbers : 50 numbers x average value = 50 x 102 = 5100 .

number of terms = 200/4 = 50

∑ = 50(a₁ + a₅₀)/2

= 50(4 + 200)/2

= 50(204)/2

= 5,100