+26367 How many numbers are in the list \(3\dfrac23,\, 4\dfrac13,\, 5,\, 5\dfrac23,\, \ldots,\, 26\dfrac13,\, 27\) ?
\(\begin{array}{|rcll|} \hline && 3\dfrac23,\, 4\dfrac13,\, 5,\, 5\dfrac23,\, \ldots,\, 26\dfrac13,\, 27 \\\\ &=& \dfrac{11}3,\, \dfrac{13}3,\, \dfrac{15}3,\, \dfrac{17}3,\, \ldots,\, \dfrac{79}3,\, \dfrac{81}3 \\\\ \text{arithmetic progression, formula: }\quad \mathbf{a_n} &=& \mathbf{a_1+(n-1)d} \\\\ && a_1 = \dfrac{11}3,\qquad a_n = \dfrac{81}3,\qquad d = \dfrac{2}3 \\\\ \dfrac{81}3 &=& \dfrac{11}3+(n-1)\dfrac{2}3 \\\\ \dfrac{81}3- \dfrac{11}3 &=& (n-1)\dfrac{2}3 \\\\ \dfrac{70}3 &=& (n-1)\dfrac{2}3 \quad | \quad *3 \\\\ 70 &=& (n-1)*2 \quad | \quad :2 \\\\ 35 &=& n-1 \\\\ n &=& 35+1 \\\\ \mathbf{n} &=& \mathbf{36} \\ \hline \end{array}\)
+26367 How many numbers are in the list \(3\dfrac23,\, 4\dfrac13,\, 5,\, 5\dfrac23,\, \ldots,\, 26\dfrac13,\, 27\) ?
\(\begin{array}{|rcll|} \hline && 3\dfrac23,\, 4\dfrac13,\, 5,\, 5\dfrac23,\, \ldots,\, 26\dfrac13,\, 27 \\\\ &=& \dfrac{11}3,\, \dfrac{13}3,\, \dfrac{15}3,\, \dfrac{17}3,\, \ldots,\, \dfrac{79}3,\, \dfrac{81}3 \\\\ \text{arithmetic progression, formula: }\quad \mathbf{a_n} &=& \mathbf{a_1+(n-1)d} \\\\ && a_1 = \dfrac{11}3,\qquad a_n = \dfrac{81}3,\qquad d = \dfrac{2}3 \\\\ \dfrac{81}3 &=& \dfrac{11}3+(n-1)\dfrac{2}3 \\\\ \dfrac{81}3- \dfrac{11}3 &=& (n-1)\dfrac{2}3 \\\\ \dfrac{70}3 &=& (n-1)\dfrac{2}3 \quad | \quad *3 \\\\ 70 &=& (n-1)*2 \quad | \quad :2 \\\\ 35 &=& n-1 \\\\ n &=& 35+1 \\\\ \mathbf{n} &=& \mathbf{36} \\ \hline \end{array}\)
