Question

1+3+...+199=
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Answer 1

1 + 3 + ... + 2n - 1 = n^2

2n - 1 = 199 => 2n = 200 => n = 100

1 + 3 + ... + 199 = 100^2 = 10 000

Answer 2

 

$\displaystyle\\\text{Aplicam formula lui Gauss:}\\\\S = \frac{\text{Numarul de termeni (Ultimul termen + primul termen)}}{2}\\\\\text{Calculam numarul de termeni:}\\\\n=\frac{\text{Ultimul termen} - \text{primul termen}}{\text{ratia progresiei aritmetice}}+1\\\\\text{Nota: Formula lui Gauss se aplica doar la numere in progresie aritmetica.}\\\\\text{Rezolvare:}\\\\n=\frac{199-1}{2}+1=\frac{198}{2}+1=99+1=100~\text{termeni}\\\\S=\frac{100(199+1)}{2}=\frac{100\times200}{2}=100\times100=100^2=10\,000\\\\\boxed{S=10\,000}$