Answer:f(n) = 12n + 1
Step-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
