Arithmetic Series Questions (Problem 1)

Number Theory Level pending

Using the formula S n S_n = n 2 \frac{n}{2} [ 2 a + ( n − 1 ) d ] [2a + (n - 1)d] , find the sum of the arithmetic series that starts from 1 and finishes at 1,000,000 (1 + 2 + ... + 999,999 + 1,000,000).


The answer is 500000500000.

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1 solution

Using S n S_n = n 2 \frac{n}{2} [ 2 a + ( n − 1 ) ∗ d ] [2a + (n - 1) * d] : a = 1, n = 1,000,000 and d = 1 so S n S_n = 1 , 000 , 000 2 \frac{1,000,000}{2} [ 2 ∗ 1 + ( 1 , 000 , 000 − 1 ) ∗ 1 ] [2 * 1 + (1,000,000 - 1) * 1] = 500,000 [ 2 ∗ 1 + ( 1 , 000 , 000 − 1 ) ∗ 1 ] [2 * 1 + (1,000,000 - 1) * 1] = 500,000 [ 2 + ( 999 , 999 ) ∗ 1 ] [2 + (999,999) * 1] = 500,000 [ 2 + ( 999 , 999 ) ] [2 + (999,999)] = 500,000 [ ( 1 , 000 , 001 ) ] [(1,000,001)] = 500,000(1,000,001) = 500,000,500,000

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