If sm =sn then what is the value of sm+n
Answers
Question:
If S(m) = S(n) , then what is the value of S(m+n).
Answer:
S(m+n) = 0
Note:
• A sequence in which, the difference between the consecutive terms are same is called AP (Arithmetic Progression).
• Any AP is given as ; a , (a + d) , (a + 2d) , .....
• The nth term of an AP is given by ;
T(n) = a + (n - 1)•d , where a is the first term and d is the common difference of the AP .
• The common difference of an AP is given by ;
d = T(n) - T(n-1) .
• The sum of first n terms of an AP is given by ;
S(n) = (n/2)•[2a + (n-1)•d] .
• The nth term of an AP is given by ;
T(n) = S(n) - S(n-1) .
Solution:
We have;
=> S(m) = S(n)
=> (n/2)•[2a + (n-1)•d] = (m/2)•[2a + (m-1)•d]
=> n•[2a + (n-1)•d] = m•[2a + (m-1)•d]
=> 2an + n(n-1)•d = 2am + m(m-1)•d
=> 2an + n(n-1)•d - 2am - m(m-1)•d = 0
=> 2an - 2am + n(n-1)•d - m(m-1)•d = 0
=> 2a•(n-m) + [n(n-1) - m(m-1)]•d = 0
=> 2a•(n-m) + [n² - n - m² + m]•d = 0
=> 2a•(n-m) + [n² - m² - n + m]•d = 0
=> 2a•(n-m) + [(n+m)(n-m) - (n-m)]•d = 0
=> 2a•(n-m) + (n-m)•[m+n - 1]•d = 0
=> (n-m)•[2a + (m+n - 1)•d] = 0
=> 2a + (m+n - 1)•d = 0 ------(1)
Now,
=> S(m+n) = [(m+n)/2]•[2a + (m+n - 1)•d]
=> S(m+n) = [(m+n)/2]•0
=> S(m+n) = 0 { using eq-(1) }
Hence,
The required value of S(m+n) is 0 .
Answer:
SM+n=0
Step-by-step explanation: