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

f(n) = 12n + 1

Step-by-step explanation:

Since it is an arithmetic sequence, the general formula for nth term is given as

f(n) = a + ( n-1)d , where a is the first term, n is the number of terms and d is the common difference.

Given from the question

Second term  is 25 , which means that

a + d = 25  

Also given , the sum of third and fifth term is 86, which means

a + 2d + a + 3d = 86

2a + 5d = 86

Combining the two equations , we have

a + d = 25 ………….. I

2a + 5d = 86………..II  

Using substitution method  to solve the resulting simultaneous equation

From equation I make a the subject of the formula, which gives

a = 25 – d…………………. III

Substitute the value of a into equation II , we have

2 ( 25 – d) + 5d = 86

Expanding

50 + 2d + 5d = 86

50 + 3d = 86

Collect the like terms

3d = 86 – 50

3d = 36

d = 12

substitute the value of d into equation III, we have

a = 25 – 12

a = 13

Since we have gotten the value of a and b , we will substitute into the general formula for the nth term

f(n) = a + ( n-1)

f(n) = 13 + (n-1)12

Expanding

f(n) = 13+ 12n -12

f(n) =12n + 1

Therefore the explicit form f(n) of the arithmetic sequence is f(n) =12n + 1