Hello, americo74!
The sum of a series is given by: \(\displaystyle \,S_n\:=\: 4n^2\,+\, 3n\)
Use the result that: \(\displaystyle \,t_{n+1}\:=\:S_{n+1}\,-\,S_n\,\) to prove that the sequence \(\displaystyle \{t_n\}\), is arithmetic.
Find both the functional and iterative equations for the sequence \(\displaystyle \{t_n\}\).
Since \(\displaystyle t_n\:=\:S_{n+1}\,-\,S_n\), we have:
\(\displaystyle \;\;t_n\;=\;\left[4(n+1)^2\,+\,3n\right] \,-\,\left[4n^2\,+\,3n\right] \;=\;8n\,+\,7\)
Hence, the sequence is: \(\displaystyle \,7,\;15,\;23,\;31,\;\cdots\;8n+7,\;\cdots\;\;\;\) for \(\displaystyle n\,=\,0,1,2,3,\cdots\)
It is an arithmetic sequence with first term \(\displaystyle t_o\,=\,7\) and common difference \(\displaystyle d\,=\,8\).
\(\displaystyle \;\;\)That is: \(\displaystyle \,t_o\,=\,7,\;t_{n+1}\:=\:t_n\,+\,8\)
