Prove that if the number ofterms of an A. P. is odd then the middle te

1.	Prove that if the number ofterms of an A. P. is odd then the middle term is the A M.between the first and last terms
Answer» i) Let the 1st term of the AP be 'a' and its common difference be 'd' ii) As the number of terms are odd, the nth term is (2n + 1) iii) Hence nth term (last tern) is: a + (2n + 1 - 1)d = a + 2nd iv) So Arithmetic mean between 1st & last term is:(a + a + 2nd)/2 = 2(a + nd)/2 = a + nd v) The middle term of (2n + 1)th term is: (2n + 1 + 1)/2 = (n + 1)Hence middle term is (n + 1)th term = a + (n + 1 - 1)d = a + nd Thus from steps (iv) & (v), Middle term = AM of 1st and last terms.

Prove that if the number ofterms of an A. P. is odd then the middle term is the A M.between the first and last terms

Answer»

i) Let the 1st term of the AP be 'a' and its common difference be 'd'

ii) As the number of terms are odd, the nth term is (2n + 1)

iii) Hence nth term (last tern) is: a + (2n + 1 - 1)d = a + 2nd

iv) So Arithmetic mean between 1st & last term is:(a + a + 2nd)/2 = 2(a + nd)/2 = a + nd

v) The middle term of (2n + 1)th term is: (2n + 1 + 1)/2 = (n + 1)Hence middle term is (n + 1)th term = a + (n + 1 - 1)d = a + nd

Thus from steps (iv) & (v),

Middle term = AM of 1st and last terms.