for a given arithmetic sequence, un=M and um=n find (a) the common difference (b) its first term

The given arithmetic progression  has the nth term un = M
and  the mthterm um = n.

To find the  (i)common difference
d and the first
term.

Solution:

The relation
between the first term a1 and the rth term ar and the common diference d  for an
arithmetic progression(AP) is given by:

ar =
a1+(r-1)d.

Now by the above relation , the formula for the
given AP, nth and mth terms could be rewriten as below:

ur
= u1+(r-1)d  becomes relation.

um = u1+(n-1)d = N
......(1)

um = u1+(m-1)d =
m.....(2)

Now solve for u1 and d in terms of n,m and
N

(1) -(2) gives:  u1+(n-1)d - u1-(m-1)d =
N-m

(n-m)d =
N-m

d = (N-m)/(n-m) is the
common difference.

The first term u1 is got by substituting
the value of d in  eq (1)(any one of the 2equations (1) or
(2)).

u1+(n-1)d = N.

u1 +
(n-1)(N-m)/(n-m) = N

 U1 = N- 
(n-1)(N-m)/(n-m)

u1 = {N(n-m)-
(n-1)(N-m)}/(n-m)

u1 = {Nn -Nm - nN +nm+N
-m}/(n-m)

u1 = {nm -Nm
+N-m}/(n-m)

u1 =  {n(m-N) -
1(m-N)}/(n-m)

u1 = (m-N)(n-1)/(n-m) is the
1st term.