Question

Escreva a P.A em que o 4° termo vale 24 e o 9° termo vale 76

Answer 1

$\mathsf{Entre~os~termos~4~e~9~h\'a~4~termos~que~podem~interpolados.}\\\\\\\mathsf{Interpola\c{c}\~ao~aritm\'etica,~descobrindo~a~raz\~ao:}\\\\\\\mathsf{r=\dfrac{b-a}{K+1}}\\\\\\\textsf{Tome b como 76, a como 24 e K como 4.}\\\\\\\mathsf{r=\dfrac{72-24}{4+1}~\Rightarrow~\dfrac{52}{5}}$


$\textsf{Agora basta descobrir o primeiro termo para montar a progress\~ao.}\\\textsf{Para isso, basta utilizar da seguinte nota\c{c}\~ao:}\\\\\\\mathsf{a_1+8r=a_9}\\\\\mathsf{a_1=a_9-8r}\\\\\\\mathsf{a_1=76-8\cdot \begin{pmatrix}\dfrac{52}{5}\end{pmatrix}}\\\\\\\mathsf{a_1=76-\dfrac{416}{5}~~~~(reduzindo~ao~mesmo~denominador)}\\\\\\\mathsf{a_1=-\dfrac{36}{5}}$


$\textsf{Seguindo a f\'ormula de recorr\^encia a progress\~ao fica da seguinte forma:}\\\\\\\mathsf{a_1+r=a_2~~...~~a_n+r=a_{n+1}}\\\\\\\mathsf{Os~ primeiros~9~termos~s\~ao:}\\\\\\\mathsf{\begin{pmatrix}-\dfrac{36}{5},\dfrac{16}{5},\dfrac{68}{5},\dfrac{120}{5},\dfrac{172}{5},\dfrac{224}{5},\dfrac{276}{5},\dfrac{328}{5},\dfrac{380}{5} \end{pmatrix}}~~~~\Large\checkmark$


$\mathsf{Observe~que~o~quarto~termo~\'e~~\dfrac{120}{5}~~que~\'e~o~mesmo~que~24,~e~o~nono}\\\\\mathsf{termo~\'e~~\dfrac{380}{5}~que~\'e~o~mesmo~que~76,~ou~seja,~a~progress\~ao~atende~aos}\\\\\mathsf{crit\'erios~da~quest\~ao.}$