Question

$\: se \: considera \: numarul \: a \: = 13 + 12 \times 13 + 12 \times {13}^{2} +12 \times {13}^{3} + ... + 12 \times {13}^{2003}. \: aratati \: ca \: a \: = 132004$
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Answer 1

$\it a=13+12\cdot13+12\cdot13^2+12\cdot13^3+\ ...\ +12\cdot13^{2003}=\\ \\ \\13+12(13+13^2+13^2+13^3+\ ...\ +13^{2003})=13+12\cdot13\cdot\dfrac{13^{2003}-1}{13-1}=\\ \\ \\ =13-13(13^{2003}-1)=13=13^{2004}-13=13^{2004}$