Răspuns:
Explicație pas cu pas:
a)
2020² - 2020 - 2019 =
= 2020 × ( 2020 - 1 ) - 2019 =
= 2020 × 2019 - 2019 =
= 2019 × ( 2020 - 1 ) =
= 2019 × 2019 =
= 2019² → patrat perfect
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b)
2 + 2¹ + 2² + 2³ + 2⁴ + 2⁵ =
= 2 + 2 + 4 + 8 + 16 + 32 =
= 64 =
= 8² → patrat perfect
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c)
1 + 3 + 5 + .... + 49 =
→ stabilesc cati termeni are suma numerelor impare
( 49 - 1 ) : 2 + 1 = 48 : 2 + 1 = 25 termeni are suma
→ aplic formula sumei lui Gauss:
= 25 × ( 1 + 49 ) : 2 =
= 25 × 50 : 2 =
= 25 × 25 =
= 25² → patrat perfect
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d)
2019 + 2 × ( 1 + 2 + 3 + ..... + 2018 ) =
= 2019 + 2 × 2018 × ( 1 + 2018) : 2 =
= 2019 + 2018 × 2019 =
= 2019 × ( 1 + 2018 ) =
= 2019 × 2019 =
= 2019² → patrat perfect
