Question

calculati:a=1+3+5...+2017 este p.p.; c=81+2•81+...+49•81 este p.p.; d=2•(1+2+3+...+124)+125 este c.p.

Answer 1

a = 1 + 3 + 5 + ..... + 2017

→ stabilesc cati termeni are suma cu ratia = 2

( 2017 - 1 ) : 2 + 1 = 2016 : 2 + 1 = 1008 + 1 = 1009 termeni

→ aplic formula sumei lui Gauss

a = 1009 × ( 1 + 2017 ) : 2

a = 1009 × 2018 : 2

a = 1009 × 1009

a = 1009²  → patrat perfect

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c = 81 + 2 × 81 + 3 × 81 + ....... + 49 × 81

c = 81 × ( 1 + 2 + 3 + ..... + 49 )

c = 81 × 49 × ( 1 + 49 ) : 2

c = 9² × 7² × 25

c = ( 9 × 7 × 5 ) ²

c = 315² → patrat perfect

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d = 2 × ( 1 + 2 + 3 + ...... + 124 ) + 125

d = 2 × 124 × ( 1 + 124 ) : 2 + 125

d = 124 × 125 + 125

d = 125 × ( 124 + 1 )

d = 125 × 125

d = 125²  → patrat perfect

d = ( 5³) ²= ( 5²) ³ → cub perfect