Math, asked by sriyanshpadhee5, 11 months ago

if mth term of an A p is eqeal to n times the nth term
find the (m+n)th term of an A.P​

Answers

Answered by bhakti59
1

formula

tn=a+(n-1)d

m term of AP

tm=a+(m-1) d...1

ń term of AP

tn=a+(n-1)d...2

from the give condition

m×tm=n×tn

m×[a+(m-1)d]=n×[a+(n-1)d]...[from 1&2]

am+m(m-1)d=an+n(n-1)d

am+m²d-md=an+n²d-nd

am+m²d-n²d-md+nd=0

a(m-n) +d(m²-n²)-d(m-n) =0

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

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

[a+(m+n-1) d]=0....3

(m+n)th term of the AP

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

t(m+n)=0.....[from 3]

Answered by sanyamshruti
0

Answer:

Let the first term of AP = a

common difference = d

We have to show that (m+n)th term is zero or a + (m+n-1)d = 0

mth term = a + (m-1)d

nth term = a + (n-1) d

Given that 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) + (m²-n²)d - (m-n)d = 0

⇒ a(m-n) + {(m-n)(m+n)}d -(m-n)d = 0

⇒ a(m-n) + {(m-n)(m+n) - (m-n)} d = 0

⇒ a(m-n)  + (m-n)(m+n -1) d  = 0

⇒ (m-n){a + (m+n-1)d} = 0 

⇒ a + (m+n -1)d = 0/(m-n)

⇒ a + (m+n -1)d = 0

Proved!

Similar questions