Question

Aratati ca A=(1•2+ 2•3+ ... +100•101)-5050 poate fi scris ca suma patratelor pentru o suta de termeni

Answer 1

A=1*2+2*3+3*4+....100*101-1-2-3-4-....-100. Grupap termenii cate 2 A=(1*2-1)+(2*3-2)+(3*4-3)+....100*101-100=1+2la puterea2+3la puterea3+......100 la puterea2

Answer 2

A=(1•2+ 2•3+ ... +100•101)-5050 

$1\cdot2=1(1+1)=1+1=1^2+1$

$2\cdot3=2(2+1)=2\cdot2+2=2^2+2$

$3\cdot4=3(3+1) = 3\cdot3+3=3^2+3$
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$100\cdot101=100(100+1)=100\cdot100+100=100^2+100$

$A=(1^2+2^2+3^2+ ... +100^2)+(1+2+3+ ... +100)-5050$

Dar, 1+2+3+ ... +100 =5050.

Deci, $A=(1^2+2^2+3^2+ ... +100^2)+5050-5050$

 $A=1^2+2^2+3^2+ ... +100^2$