Summing and subtracting

Algebra Level 2

1 − 2 + 3 − 4 + ⋯ + 999 − 1000 + 1001 = ? \large1-2+3-4+\cdots+999-1000+1001=\, ?


The answer is 501.

Tarmo Taipale by Tarmo Taipale , 22, Finland

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2 solutions

Tarmo Taipale
Oct 8, 2016

1 − 2 + 3 − 4 + . . . + 999 − 1000 + 1001 1-2+3-4+...+999-1000+1001

= ( 1 − 2 ) + ( 3 − 4 ) + . . . + ( 999 − 1000 ) + 1001 =(1-2)+(3-4)+...+(999-1000)+1001

= 500 × ( − 1 ) + 1001 =500\times(-1)+1001

= 501 =\boxed{501}

9 Helpful 0 Interesting 0 Brilliant 0 Confused

well my process was bit lengthy,I solved it using properties of arithmetic progressions. summing negative and positive integers and then adding them.

Ishaan Shanker - 4 years, 8 months ago

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That's another way to solve this problem. :)

Tarmo Taipale - 4 years, 8 months ago
Viki Zeta
Oct 8, 2016

S = 1 − 2 + 3 − 4 + … + 999 − 1000 + 1001 S = ( 1 + 3 + 5 + 7 + … + 1001 ) − ( 2 + 4 + 6 + 8 + … + 1000 ) S = S a + S b , where S a and S b are sum to n terms of AP S = 501 2 ( 2 ( 1 ) + ( 501 − 1 ) ( 2 ) ) − 500 2 ( 2 ( 2 ) + ( 500 − 1 ) 2 ) = 501 2 ( 2 + 1000 ) − 500 2 ( 4 + 998 ) = 501 2 ( 2 + 1000 ) − 500 2 ( 2 + 1000 ) = ( 2 + 1000 ) ( 501 2 − 500 2 ) = ( 2 + 1000 ) ( 500 + 1 2 − 500 2 ) = ( 2 + 1000 ) ( 500 2 + 1 2 − 500 2 ) = ( 2 + 1000 ) ( 1 2 ) = 1 + 500 = 501 S = 1 - 2 + 3 - 4 + \ldots + 999 - 1000 + 1001 \\ S = \color{#3D99F6}{(1+3+5+7+\ldots + 1001)} - \color{#D61F06}{(2+4+6+8+\ldots+1000)} \\ S = S_a + S_b \text{, where }S_a\text{ and } S_b \text{ are sum to n terms of AP} \\ S = \color{#3D99F6}{\dfrac{501}{2}(2(1) + (501-1)(2))} - \color{#D61F06}{\dfrac{500}{2}(2(2) + (500-1)2)} \\ = \dfrac{501}{2}(2 + 1000) - \dfrac{500}{2}(\color{#D61F06}{4} + 998) \\ = \dfrac{501}{2}(2+1000) - \dfrac{500}{2}(\color{#D61F06}{2} + \color{#3D99F6}{1000}) \\ = \color{#3D99F6}{(2+1000)}\left(\dfrac{\color{#D61F06}{501}}{2} - \dfrac{500}{2}\right) \\ = (2+1000)\left(\dfrac{500 + 1}{2} - \dfrac{500}{2}\right) \\ = (2+1000)\left(\color{#D61F06}{\dfrac{500}{2}} + \dfrac{1}{2} - \color{#D61F06}{\dfrac{500}{2}}\right) \\ = (2+1000)\left(\dfrac{1}{2}\right) \\ = 1 + 500 \\ \boxed{= 501}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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