Find the value of the middle term of the following A.P.: -6, -2, 2, …, 58.

Or

Determine the A.P. whose fourth term is 18 and the difference of the ninth term from the fifteenth term is 30.

nth term of an A.P. is given by a_n=a + (n-1)d

Given: a=-6; d=-2-(-6) =4; a_n=58

Substituting the values of a and d in the formula of a_n

a + (n-1)d=58

⇒ (-6) + (n-1)4=58

⇒(n-1)4=58 + 8=64
⇒n-1=64/4=16

⇒n=17

We see that n is odd.

So,Middle term=th term==9^th term

Middle term = a + (n-1)d =( -6) + (9-1)4=(-6) + 32=26

Or

Let us assume that,

First term of A.P.=a

Common difference=d

We know that nth term of an A.P. is given by a_n=a + (n-1)d

4^th term of A.P.= a + 3d =18(given)…….(1)

Also, 15^th term – 9^th term =30

⇒ (a + 14d)-(a + 8d) =30

⇒ 6d =30

⇒ d=5

Putting the value of d in equation 1

a + 3(5) =18

a + 15= 18

a= 3

So the A.P. is 3,8,13,18,23,28,33……..