Question

Sea la progresión aritmética: 7, 12, 17, 22, ..., Calcule la diferencia entre los términos de lugares 20 y 10.

Answer 1

Recordemos que el término general en una progresión aritmética

$\boxed{\sf{a_n=a_1+(n-1)r}}\hspace{20pt} \mathsf{Donde} \hspace{15pt}\begin{array}{ll}\mathsf{\blue{\rightarrow}\ a_n:T\acute{e}rmino\ n-\acute{e}simo} &\mathsf{\blue{\rightarrow}\ r:Raz\acute{o}n}\\\\\mathsf{\blue{\rightarrow}\ a_1:Primer\ t\acute{e}rmino}&\sf{\blue{\rightarrow}\ n:N^{\circ}\ de\ t\acute{e}rminos}\end{array}$

Partimos de lo que nos piden,

            $\begin{array}{cccccccccccc}\begin{array}{c}\underline{\boldsymbol{\sf{T\acute{e}rmino\ 10}}}\\\\\sf{a_{n}=a_1 + (n-1)r}\\\\\sf{a_{10}=a_1 + (10-1)r}\\\\\boxed{\sf{a_{10}=a_1 + 9r}}\end{array}&&&&&&&&\begin{array}{c}\underline{\boldsymbol{\sf{T\acute{e}rmino\ 20}}}\\\\\sf{a_{n}=a_1 + (n-1)r}\\\\\sf{a_{20}=a_1 + (20-1)r}\\\\\boxed{\sf{a_{20}=a_1 + 19r}}\end{array}\end{array}$

La diferencia será

                        $\begin{array}{l}\sf{T\acute{e}rmino\ 20 - T\acute{e}rmino\ 10 = a_{20} - a_{10}}\\\\\sf{T\acute{e}rmino\ 20 - T\acute{e}rmino\ 10 = (a_1 + 19r) - (a_1 + 9r)}\\\\\sf{T\acute{e}rmino\ 20 - T\acute{e}rmino\ 10 = a_1 + 19r -a_1 - 9r}\\\\\sf{T\acute{e}rmino\ 20 - T\acute{e}rmino\ 10 = 10r}\\\end{array}$

Para determinar su diferencia solo necesitamos hallar la razón, y esta la obtendremos de la progresión:

                                       $\begin{array}{l} \sf{\overset{\displaystyle\overset{a_1}{\downarrow }}{7}\quad,\quad \overset{\displaystyle\overset{a_2}{\downarrow }}{12}\quad,\quad \overset{\displaystyle\overset{a_3}{\downarrow }}{17}\quad,\quad \overset{\displaystyle\overset{a_4}{\downarrow }}{22}\quad,\quad \cdots\quad ,}\\\hspace{14pt}\overset{\displaystyle\smile}{5}\hspace{24pt}\overset{\displaystyle\smile}{5}\hspace{24pt}\overset{\displaystyle\smile}{5}\end{array}\\\\\\ \Longrightarrow\ \boxed{\sf{r=5}}$

Reemplazamos "r" para hallar la diferencia

                        $\begin{array}{l}\sf{T\acute{e}rmino\ 20 - T\acute{e}rmino\ 10 = 10r}\\\\\sf{T\acute{e}rmino\ 20 - T\acute{e}rmino\ 10 = 10(5)}\\\\\boxed{\boxed{\boldsymbol{\sf{T\acute{e}rmino\ 20 - T\acute{e}rmino\ 10 = 50}}}}\end{array}$

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