if m times the nth term of an AP is equal to n times it's nth term show that it's (m+ n)th term is zero
Answers
Answer:
0
Step-by-step explanation:
let the first be and common difference is .
Given m times the mth term = n times the nth term
Recall that the nth term of AP is
Similarly, the mth term of AP is
Step-by-step explanation:
Given Question:-
If m times the nth term of an AP is equal to n times it's nth term show that it's
(m+ n)th term is zero.
Correct Question:-
If m times the mth term of an AP is equal to n times it's nth term show that it's
(m+ n)th term is zero.
Given:-
m times the mth term of an AP is equal to n times it's nth term .
To find:-
Show that it's (m+ n)th term is zero.
Solution:-
We know that
a is the first term and d is the common difference of an AP then nth term of the AP is a+(n-1)d
Given that
m times the mth term of an AP
=> m × tm
=> m× [a+(m-1)d]------(1)
And
n times the nth term of the AP
=> n×tn
=> n×[a+(n-1)d]------(2)
According to the given problem
m tm = n tn
=> m× [a+(m-1)d]=n×[a+(n-1)d]
=> m [a+md-d] = n×[a+nd-d]
=> am + m^2d -dm = an +n^2d -nd
=> am + m^2d -dm -an -n^2d +nd=0
=> (am-an)+(m^2d-n^2d)+(nd-dm)=0
=>a(m-n)+d(m^2-n^2)+d(n-m) = 0
=>a(m-n)+d(m+n)(m-n)-d(m-n)=0
=>a(m-n)+[d(m-n)(m+n-1)]=0
=> (m-n)[a+(m+n-1)d]=0
=> m-n = 0 or a+(m+n-1)d=0
=> m =n or a+(m+n-1)d=0
a+(m+n-1)d=0
(m+n)th term = 0
=> t (m+n) = 0
Answer:-
If m times the mth term of an AP is equal to n times it's nth term show that it's (m+ n)th term is zero.
Used formulae:-
- a is the first term and d is the common difference of an AP then nth term of the AP is a+(n-1)d
- (a+b)(a-b)=a^2-b^2