Question

aratati ca nr d= 2x(1+2+3+....+124) +125 este cub perfect

Answer 1

d = 2 × (1 + 2 + 3 + ... + 124) + 125

d = 2 × 124 × (124 + 1) : 2 + 125

d = 2 × 124 × 125 : 2 + 124

am folosit suma lui Gauss

2: 2 = 1 si de aceea se reduc

d = 124 × 125 + 125

dam factor comun pe 125

d = 125 × ( 124 + 1)

d = 125 × 125

d = 125^2

d = (5^3)^2

d = (5^2)^3

d = 25^3 cub perfect
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Answer 2

d= 2·(1+2+3+....+124) +125

(1+2+3+....+124)  suma Gauss cu formula de calcul S= [n(n+1):2= (124 · 125):2=

deci d devine : d =2·(125 ·124):2 +125 = 125·124 +125 acum dam factor comun 125

d= 125 (124 + 1) =125 · 125 = 5³ · 5³ = 5³⁺³ = 5⁶ = (5²)³ = 25³