Menu Close

In-an-AP-if-the-m-th-term-is-n-and-the-n-th-term-is-m-where-m-is-not-equal-to-n-find-1-The-value-of-m-n-th-term-2-The-p-th-term-

Tinku Tara 0 Comments
Pin




Question Number 7399 by Tawakalitu. last updated on 27/Aug/16
In an AP , if the m^(th) term is n and the n^(th) term is m. where m is not equal to n. find (1) The value of (m + n)^(th) term (2) The p^(th) term.
InanAP,ifthemthtermisnandthenthtermism.wheremisnotequalton.find(1)Thevalueof(m+n)thterm(2)Thepthterm.
Answered by Yozzia last updated on 27/Aug/16
For an AP, the k^(th) term u(k) is given by u(k)=u(1)+(k−1)d where d is a constant. ∴ k=m⇒u(m)=u(1)+(m−1)d=n and k=n⇒u(n)=u(1)+(n−1)d=m (1) k=m+n⇒u(m+n)=u(1)+(m+n−1)d But, n−m=(m−n)d and m≠n⇒d=−1 ∴u(m+n)=u(1)+(m+n−1)(−1) but, n+m=2u(1)+(−1)(m+n−2) 2(m+n)=2u(1)+2 u(1)=m+n−1 ∴ u(m+n)=m+n−1+(−1)(m+n−1)=0 (2) k=p⇒ u(p)=m+n−1+(p−1)(−1)=m+n−p
ForanAP,thekthtermu(k)isgivenbyu(k)=u(1)+(k−1)dwheredisaconstant.∴k=m⇒u(m)=u(1)+(m−1)d=nandk=n⇒u(n)=u(1)+(n−1)d=m(1)k=m+n⇒u(m+n)=u(1)+(m+n−1)dBut,n−m=(m−n)dandm≠n⇒d=−1∴u(m+n)=u(1)+(m+n−1)(−1)but,n+m=2u(1)+(−1)(m+n−2)2(m+n)=2u(1)+2u(1)=m+n−1∴u(m+n)=m+n−1+(−1)(m+n−1)=0(2)k=p⇒u(p)=m+n−1+(p−1)(−1)=m+n−p
Commented by Rasheed Soomro last updated on 27/Aug/16
Another way From the above d=−1 u(1)+(m−1)d=n adding nd to both sides u(1)+(m−1)d+nd=n+nd u(1)+(m+n−1)d=n+n(−1) u(1)+(m+n−1)d=0 u(m+n)=0 u(1)+(m−1)d=n Subtracting (m−p)d to both sides u(1)+(m−1)d−(m−p)d=n−(m−p)d u(1)+(m−1−m+p)d=n−(m−p)(−1) u(1)+(p−1)d=n+m−p u(p)=m+n−p
AnotherwayFromtheaboved=−1u(1)+(m−1)d=naddingndtobothsidesu(1)+(m−1)d+nd=n+ndu(1)+(m+n−1)d=n+n(−1)u(1)+(m+n−1)d=0u(m+n)=0u(1)+(m−1)d=nSubtracting(m−p)dtobothsidesu(1)+(m−1)d−(m−p)d=n−(m−p)du(1)+(m−1−m+p)d=n−(m−p)(−1)u(1)+(p−1)d=n+m−pu(p)=m+n−p
Commented by Tawakalitu. last updated on 27/Aug/16
Thank you sir. i really appreciate
Thankyousir.ireallyappreciate

Pin

Leave a Reply

Your email address will not be published. Required fields are marked *