Question

arata ca urmatoarele numere sunt patratele unor numere naturale: c) 1+3+5+......+99

Answer 1

1 + 3 + 5 + ...... + 99 =

= 1 +  ( 1 + 2 ) + ( 1 + 4 ) + ( 1 + 6 ) + ...... + ( 1 + 98 ) =

= 1 × 50 + 2 × ( 1 + 2 + 3 + ..... + 49 ) =

= 50 + 2 × 49 × ( 1 + 49 ) / 2 =

= 50 + 49 × 50 =

= 50 × ( 1 + 49 ) =

= 50 × 50 =

= 50² → patrat perfect

_______________________

S = 1 + 3 + 5 + ......... + 99

→ stabilesc cati termeni are suma cu ratia = 2

( 99 - 1 ) : 2 + 1 = 98 : 2 + 1 = 50 termeni

→ aplic formula sumei lui Gauss

S = 50 × ( 1 + 99 ) : 2

S = 50 × 100 : 2

S = 5000 : 2

S = 2500 → patratul numarului 50

S = 50²