Question

BEST GETS BRAINLIEST :
If m times the mth term of an A.P. = n times the nth term then show that the (m+n)th term of the A.P. is zero

Answer 1

m(am)=n(an)
m(a+(m-1)d)=n(a+(n-1)d)
am+m^2d-md=an+n^2d-nd
am+m^2d-md - an -n^2d+nd=0
am-an + m^2d - n^2d - md + nd=0
a(m-n) + d(m^2 - n^2) -d(m - n)=0
a(m-n) + d(m-n)(m+n) -d(m - n)=0
(m-n)(a+d(m+n)-d)=(m-n)×0
a+d(m+n)-d=0
a+d(m+n-1)=0
a+(m-n+1)d=0
a(m+n)=0
hence proved

Answer 2

The answer is given below :

Let us consider that the first term of the AP is a and the common difference is d.

Then, the mth term

= a + (m - 1)d

and

the nth term

= a + (n - 1)d

Given that,

m times the mth term = n times the nth term

⇒ m × [a + (m - 1)d] = n × [a + (n - 1)d]

⇒ ma + m(m - 1)d = na + n(n - 1)d

⇒ ma + m²d - md = na + n²d - nd

⇒ (m - n)a + (m² - n²)d - (m - n)d = 0

⇒ (m - n)a + (m + n)(m - n)d - (m - n)d = 0

⇒ a + (m + n)d - d = 0, since (m - n) ≠ 0

⇒ a + (m + n - 1)d = 0 .....(i)

So, the (m + n)th term

= a + (m + n - 1)d

= 0, by (i) [Proved]

Thank you for your question.