if n times the nth term of an AP is equal to m times the mth term of AP then prove that (m+n)th term is equal to zero.
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Answered by
38
Tn=a+(n-1)d
Tm=a+(m-1)d
Also,n.Tn=m.Tm
=>n[a+(n-1)d]=m[a+(m-1)d]
=>an+n^2 d-nd=am+m^2 d-md
=>a(n-m)+(m-n)d+(n^2-m^2)d=0.
=>a(n-m)+d[(n+m)(n-m)-(n-m)]=0
=>a(n-m)+d(n-m)(n+m-1)=0
=>(n-m)[a+(m+n-1)d]=0
=>a+(m+n-1)d=0
=>T(m+n)=0
Hope it helps
Tm=a+(m-1)d
Also,n.Tn=m.Tm
=>n[a+(n-1)d]=m[a+(m-1)d]
=>an+n^2 d-nd=am+m^2 d-md
=>a(n-m)+(m-n)d+(n^2-m^2)d=0.
=>a(n-m)+d[(n+m)(n-m)-(n-m)]=0
=>a(n-m)+d(n-m)(n+m-1)=0
=>(n-m)[a+(m+n-1)d]=0
=>a+(m+n-1)d=0
=>T(m+n)=0
Hope it helps
Answered by
14
hope you get help from this
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