Question

Hi! solve guys❤Quēstion here ;
If m times the m^th term of an A.P. is equal to n times its n^th term,show that the(m+n)^th is zero(0) ⠀⠀⠀⠀


⠀Kindly solve fully :)​

Answer 1

Step-by-step explanation:

Given:

mth times mth term of AP = n times the nth term.

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

⇒ m * [a + (m - 1) * d] - n * [a + (n - 1) * d] = 0

⇒ ma + m(m - 1) * d - na - n(n - 1) * d = 0

⇒ ma + (m² - m) * d - na - (n² - n) * d = 0

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

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

⇒ (m - n)[a + d((m + n) - 1)] = 0

⇒ a + [(m + n) - 1]d = 0

Therefore,

It denotes (m + n)th  term is 0.

Hope it helps!

Answer 2

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$\huge\boxed{ANSWER}$

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M'th term

$a_m \: = \: a + (m - 1)d$

N'th term

$a_n \: = \: a + (n - 1)d$

To prove :

a + (m+n-1)d = 0

According to question

m(M'th term) = n(N'th term)

m(a + (m-1)d) = n(a + (n-1)d)

m(a+md-d) = n(a+nd-d)

ma + m²d - md = na + n²d - nd

ma - na + m²d - n²d - md + nd = 0

a(m-n) + d(m²-n²) - d(m-n) = 0

Dividing (m-n) on bhs

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

a + d(m+n-1) = 0

(m+n)th term = 0

Hence proved ✔✔

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Ab mai itna b khaas ni :shy ❤