Question

If m times the mth term of an AP is equal to n times nth term , show that (m+n)th term of the AP is zero.​

Answer 1

Step-by-step explanation:

Given:-

• m times the mth term is equal to n times the nth term of an AP.

To Prove:-

• (m + n)th term of the AP is zero.

Solution:-

Let a be the first term and d be the common difference.

As we know that:-

nth term of an AP is:-

$\dashrightarrow \rm a_{n} = a + ( n - 1)d$

Given, m times mth term = n times nth term

$\dashrightarrow \rm m(a_{m} )= n( a_{n})$

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

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

→ am + m(m - 1)d - an + n(n - 1)d = 0

→ am - an + m(m - 1)d - n(n - 1)d = 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 { m² - n² + m + n} = 0

→ a(m - n) + d { m² - n² - (m + n - 1)} = 0

→ a(m - n) + d(m - n) (m + n - 1)= 0

→ (m - n) { a + (m + n - 1)d} = 0

→ a + (m + n - 1)d= 0

$\dashrightarrow \rm a_{m + n} = 0$

Therefore, (m + n)th term of the AP is zero

Answer 2

Answer in above attachment