Question

Find the middle term of an A. P. 20,16,12,.....,-180.

Answer 1

$\huge{\textbf{\underline{\red{Solution is attached here.}}}}$

Answer 2

Given,

AP = 20,16,12,.....,-180

To Find,

The middle term of AP =?

Solution,

a(First term) in the AP = 20

Common difference(d) = a2 -a1 = 16 - 20

Common difference(d) = -4

Last term of AP = (-180)

Using formula for nth term in the AP, we have

$a_n = a + (n - 1)d$

For last term we have,

$a_n = (- 180) = 20 + (n - 1)(- 4)\\(-180 - 20) = (n - 1)(- 4)\\(- 200 ) = (n - 1)(- 4)\\(n - 1) = (-200) / ( - 4)\\(n - 1) = 200 / 4\\(n - 1) = 50\\n = 50 + 1\\n = 51$

There are 51 terms in the AP.

Middle number  of the AP = (n+1)/2 = (51 + 1 )/ 2

Middle number  of the AP = 52 / 2

Middle number  of the AP  = 26

The 26th term of AP,

$a_{26} = 20 + (26 - 1)(-4)\\a_{26} = 20 - (25 * 4\\a_{26} = 20 - 100\\a_{26} = - 80$

Hence, the middle term of an A. P. 20,16,12,.....,-180 is -80.