Question

aratati ca x e nr natural patrat perfect:

Answer 1

a)

x = ( 1 + 2 + 3 + ...... + 98 ) + 49

x = 98 × ( 1 + 98 ) : 2 + 49

x = 49 × 99  + 49

x = 49 × ( 99 + 1 )

x = 7² x 10²

x = ( 7 × 10 )² = 70² → patrat perfect

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b)  

x = 1 + 3 + 5 + ....... + 225

→ stabilesc cati termeni are suma cu ratiia 2

( 225 - 1 ) : 2 + 1 = 224 : 2 + 1 = 112 + 1 = 113  termeni are suma

→ aplic formula sumei lui Gauss

x = 113 × ( 1 + 225 ) : 2

x = 113 × 226 : 2

x = 113 × 113 = 113²  → patrat perfect

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c)

x = 1 + 3 + 5 + ............ + 2015

( 2015 - 1 ) : 2 + 1 = 2014 : 2 + 1 = 1007 + 1 = 1008 termeni are suma

x = 1008 × ( 1 + 2015 ) : 2

x = 1008 × 2016 : 2

x = 1008 × 1008  = 1008²  →  patrat perfect

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d)

x = 2 + 4 + 6 + ...... + 2016 + 1009

x = 2 × ( 1 + 2 + 3 + ..... + 1008 ) + 1009

x = 2 × 1008 × ( 1 + 1008) : 2 + 1009

x = 1008 × 1009 + 1009

x = 1009 × ( 1008 + 1 )

x = 1009 × 1009 = 1009²  → patrat perfect