Explanation
From the statement, we have an arithmetic sequence with:
• common difference d = 4,
,
• 3rd term a₃ = 19.
We must find the 77th term.
(1) The general formula for the nth term of an arithmetic sequence is:
[tex]a_n=a_1+(n-1)\cdot d.[/tex]
Replacing the data from above, we have:
[tex]a_n=a_1+4(n-1).[/tex]
(2) We compute the a₁. Replacing the n = 3 and a₃ = 19, we have:
[tex]\begin{gathered} a_3=a_1+4(3-1), \\ 19=a_1+8. \end{gathered}[/tex]
Solving for a₁, we get:
[tex]a_1=19-8=11.[/tex]
Replacing this value in the general formula, we have:
[tex]a_n=11+4(n-1).[/tex]
(3) Evaluating the general formula for n = 77, we get the 77th term of the sequence:
[tex]a_{77}=11+4\cdot(77-1)=11+304=315.[/tex]Answer
a₇₇ = 315
