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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