algebra and trigonometry

Question

The nth term of an arithmetical progression is n and the nth term is m. Find the first term and the common difference.

Answer 1

did you mean nth term is n and mth term is m. if so, check the solution here:

 

since its an arithmetic progression:

 

nth Term can be written as Tn = a1 + (n-1)d and mth term can be written as Tm = a1 + (m-1)d 

 

therefore: a1 + (n-1)d = n    -----(1)                     [its given nth term is n]

 

similarly a1+ (m-1) d = m    -----(2)                      [its given mth term is m]

 

from (1) and (2) separate a1 

 

from (1)   a1 = n - (n-1)d      -----(3)

from (2)  a1 = m - (m-1)d    ------(4)

 

from (3) and (4) they should be equal (both give value of a1

 

therefore,     n - (n-1)d  = m - (m-1)d 

 

solving,

 

n-m = (n-1)d - (m-1)d

 

=> n-m = d ( n-1-m+1)              [taking out d as a common]

=> n-m = d (n-m)

 

=> d = n-m  = 1

           n-m

 

therefore Common Difference d = 1

 

Putting this value back into (1) or (2)

 

a1 + (n-1)d = n

 

=> a1 = n - (n-1)d 

          = n - (n-1) * 1            [common difference d = 1]

         = n - n + 1

         = 1

 

Therefore, 1st term and common difference is 1.

 

 

Please let me know if this helped you. let me know if you find any mistakes in the solution. I am open to discuss.