Find an explicit form f(n) of the arithmetic sequence where the 2nd term is 25 and the sum of the 3rd term and 4thterm is 86.

Answer:f(n) = 12n + 1Step-by-step explanation:Since it is an arithmetic sequence, the general formula for nth term is given as [tex]\\[/tex]f(n) = a + ( n-1)d , where a is the first term, n is the number of terms and d is the common difference. [tex]\\[/tex]Given from the question [tex]\\[/tex]Second term  is 25 , which means that a + d = 25  [tex]\\[/tex]Also given , the sum of third and fifth term is 86, which means [tex]\\[/tex]a + 2d + a + 3d = 86 [tex]\\[/tex]2a + 5d = 86 [tex]\\[/tex]Combining the two equations , we have [tex]\\[/tex]a + d = 25 ………….. I [tex]\\[/tex]2a + 5d = 86………..II  [tex]\\[/tex]Using substitution method  to solve the resulting simultaneous equation[tex]\\[/tex]From equation I make a the subject of the formula, which gives [tex]\\[/tex] a = 25 – d…………………. III [tex]\\[/tex]Substitute the value of a into equation II , we have [tex]\\[/tex]2 ( 25 – d) + 5d = 86 [tex]\\[/tex]Expanding [tex]\\[/tex]50 + 2d + 5d = 86 [tex]\\[/tex]50 + 3d = 86 [tex]\\[/tex]Collect the like terms [tex]\\[/tex]3d = 86 – 50 [tex]\\[/tex]3d = 36 [tex]\\[/tex]d = 12 [tex]\\[/tex]substitute the value of d into equation III, we have [tex]\\[/tex]a = 25 – 12 [tex]\\[/tex]a = 13 [tex]\\[/tex]Since we have gotten the value of a and b , we will substitute into the general formula for the nth term [tex]\\[/tex]f(n) = a + ( n-1) [tex]\\[/tex]f(n) = 13 + (n-1)12 [tex]\\[/tex]Expanding [tex]\\[/tex]f(n) = 13+ 12n -12 [tex][/tex]f(n) =12n + 1 [tex]\\[/tex]Therefore the explicit form f(n) of the arithmetic sequence is f(n) =12n + 1