Question

If m times the mth term of an A.P is equal to n and nth term then show that the (m+n)th term of the A.P. is zero .​

Answer 1

Correct Question :-

If m times the mth term of an A.P is equal to n times the nth term. Show that the (m+n)th term of the A.P. is zero

Given :-

• m times mth term = n times nth term

To Prove :-

• The (m + n)th term of the AP is zero.

Solution :-

We know, nth term of an AP is of the form,

• a + (n - 1)d

Where,

• a = First term, d = Common difference
• n = Integer

According to the question:

⇒ m × (mth term) = n × (nth term)

⇒ m { a + (m - 1)d } = n { a + (n - 1)d }

⇒ ma + (m - 1)md - na - (n - 1)nd = 0

⇒ ma - na + (m - 1)md - (n - 1)nd = 0

⇒ a(m - n) + d { m(m - 1) - n(n - 1) } = 0

⇒ a(m - n) + d(m² - m - n² + n) = 0

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

⇒ a(m - n) + (m - n)d { (m + n) - 1 } = 0

⇒ (m - n) { a + (m + n - 1)d } = 0

⇒ a + (m + n - 1)d = 0

As discussed earlier, nth term of an AP is of the form a + (n - 1)d. Here, n = m + n

Which means, (m + n)th term is zero.

Hence, Proved!

Some Information :-

• An AP is a sequence of numbers where the difference between any two consecutive numbers is the same which is knowns as Common difference.

Answer 2

Answer:

Question :-

• If m times the mth term of an A.P is equal to n and nth team then show that the (m + n)th term of the A.P is zero.

Given :-

• If m times the mth term of an A.P is equal to n and nth term.

Show That :-

• Then show that the (m + n)th term of the A.P

Solution :-

➻ m (a + (m - 1)d) = n (a + (n - 1)d)

➻ am + m²d - md = an + n²d - nd

➻ am - an + m²d - n²d - md + nd = 0

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

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

➻ m - n{a + d(m + n - 1) } = 0

Now, rejecting the non - trivial case of m - n, we assume that m and n are different.

➠ {a + d(m + n - 1) } = 0

∴ The LHS of the above equation denotes (m + n)th term of the AP, which is zero.