Arati ca urmatoarele numere sunt patrate perfecte a) 17^28 / b) 25^19 / c) 1+3+5+7...+55 / Va rog repede

Răspuns

Explicație pas cu pas:

a)   17²⁸ =  17 ¹⁴ ˣ ²  = ( 17¹⁴ )² → patrat perfect

b)  25¹⁹ = ( 5² )¹⁹ = ( 5¹⁹ )² → patrat perfect

c)  1 + 3 + 5 + 7 + ..... + 55 = 28²

→ stabilesc cati termeni are suma numerelor impare

( 55 - 1 ) : 2 + 1 = 54 : 2 + 1 = 28 termeni are suma

→ aplic formula sumei lui Gauss

= 28 × ( 1 + 55 ) : 2 =

= 28 × 56 : 2 =

= 28 × 28 =

= 28² → patrat perfect

A)= 17²⁸ = (17¹⁴)² = patrat perfect

B) = 25¹⁹ = (5¹⁹)² = patrat perfect

C)= 1+3+5+.....+55=

= (1+2+3+4+......+55)-(2+4+6+......+54)= factori comun 2

= (1+2+3+...+55) - 2×(1+2+3+....+27) =

= (55×56/2) - 2×( 27×28/2) =

= 55×28 − 27×28 = 28 factor comun

28×( 55-27)=

28×28= 28² patrat perfect