Question

Aratati ca urmoatarele numere sunt patrate perfecte
a) x = 6+12+18+...+288
b) x = 4+8+12+...+196
c) x = 1+3+5+...+1001
d) x= 1+3+5+...+2001
e) x = 1203+2+4+...+2404

Vreau si eu sa stiu cum ati calculat va rog, ms mult

Answer 1

Răspuns:

Explicație pas cu pas:

a)

x = 6+12+18+....+288

x = 6×(1+2+3+....+48)

x = 6×48×(1+48) : 2

x = 3 × 48 × 49

x = 144 × 49

x = 12² × 7²

x = (12 × 7 )²

x = 84² → patrat perfect

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b)

x = 4+8+12+....+196

x = 4 × (1+2+3+....+49)

x = 4 × 49 × (1+49):2

x = 2 × 49 × 50

x = 49 × 100

x = 7² × 10²

x = 70² → patrat perfect

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c)

x = 1+3+5+...+1001

(1001-1):2+1 = 1000:2+1 = 501 termeni

x = 501 × ( 1 + 1001 ) : 2

x = 501 × 1002:2

x = 501 × 501

x = 501² → patrat perfect

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d)

x = 1+3+5+ ......+ 2001

x = [(2001+1) : 2 ]²

x = (2002/2)²

x = 1001² ->  patrat perfect

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Sau:

(2001-1):2+1 = 2000:2+1 = 1001 termeni are suma

x = 1001 × 2002:2

x = 1001× 1001

x = 1001²  -->  patrat perfect

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e)

x = 1 203 + 2 × ( 1 + 2 + .... + 1 202)

x = 1 203 + 2 × 1 202 × ( 1 + 1202) : 2

x = 1 203 + 1 202 × 1 203

x = 1203 × ( 1 + 1 202 )

x = 1 203 × 1 203

x = 1203² → patrat perfect