Solve this without calculator!

Algebra Level 3

Without using a calculator, find the value of

100 0 3 + 87 6 2 − ( 99 9 3 + 87 5 2 ) \large 1000^3+876^2-\left(999^3+875^2\right)


The answer is 2998752.

Vaibhav Priyadarshi by Vaibhav Priyadarshi , 17, India

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2 solutions

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Chew-Seong Cheong
Mar 20, 2018

x = 100 0 3 + 87 6 2 − ( 99 9 3 + 87 5 2 ) Let a = 1000 and b = 875 = a 3 + ( b + 1 ) 2 − ( a − 1 ) 3 − b 2 = a 3 + b 2 + 2 b + 1 − a 3 + 3 a 2 − 3 a + 1 − b 2 = 2 b + 2 + 3 a 2 − 3 a = 1750 + 2 + 3000000 − 3000 = 1752 + 2997000 = 2998752 \begin{aligned} x & = 1000^3+876^2-(999^3+875^2) & \small \color{#3D99F6} \text{Let }a=1000 \text{ and }b=875 \\ & = a^3 + (b+1)^2 - (a-1)^3-b^2 \\ & = a^3 + b^2+2b+1 - a^3+3a^2-3a+1-b^2 \\ & = 2b+2+3a^2-3a \\ & = 1750+2+3000000-3000 \\ & = 1752 + 2997000 \\ & = \boxed{2998752} \end{aligned}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

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