Prove that if the number of terms of an A.P is odd then the middle term is the A.M between the first and last term???
<em><u>Answer:</u></em>
Proved that when number of terms of an A.P is odd then the middle term is the A .M between the first and last term
<em><u>Solution:</u></em>
To prove: If the number of terms of an A.P is odd then the middle term is the A .M between the first and last term
Proof:
Let the arithmetic progression be:
a, a + d, a + 2d, ........., (2n + 1) terms
As the number of terms are odd, the nth term is (2n + 1)
The middle term of (2n + 1)th term is:

Hence middle term is (n + 1)th term
The formula for nth term of arithmetic progression is:
------- (i)
Where n is the terms location and "d" is the common difference between terms
Thus middle term is: (n + 1)th term
From (i)
----- (ii)
Arithmetic mean between first and last term:
The nth term is (2n + 1)
Hence nth term (last term) is:
From (i)

So Arithmetic mean between 1st & last term is:
First term of A.P = a
Arithmetic mean between first and last term = (first term + last term)/2
----- (iii)
Thus from steps (ii) and (iii)
Middle term = AM of 1st and last terms [Proved]
Thus proved that when number of terms of an A.P is odd then the middle term is the A .M between the first and last term