Question

Consider an arithmetic sequence whose m^{​​​​​​th} term is 'n' and n^​​th term is 'm'

a) Find the common difference of the sequence

b) Prove that ( m+n+p) ^th term of the sequence is '-p'

Answer 1

Solution:

Let 1st term be 'a' and common difference be 'd'.

If mth term of an Arithmetic Sequence is 'n',
mth term: a + (m-1)d = n ....i)

If nth term of an Arithmetic Sequence is 'm',
nth term: a + (n-1)d = m ....ii)

Subtracting (i) from (ii) gives:
a + (n - 1)d - a - (m-1)d = m - n

or, a + nd - d - a - md + d = m - n

or, nd - md = m-n

or, d(n - m) = m-n

or, d = (m-n)/(n-m)

or, d = -1

Ans(a): Therefore, the common difference of the sequence is = -1.

Now putting value of 'd= -1' in equation (i), we get value of 'a';

or, a + (m - 1)d = n ....i)
or, a + (m - 1)(-1) = n
or, a - m + 1 = n
or, a = m + n - 1

Therefore, (m + n + p)th term of the sequence;

(m + n + p)th term= a + (m + n + p - 1)d ....iii)

Now, putting value of 'a' & 'd' in Eq (iii);

(m + n + p)th term = a + (m + n + p - 1)d
= m + n - 1 + (m + n + p - 1)(-1)
= m + n - 1 - m - n - p + 1

Ans(b): Therefore, (m + n + p)th term of the Sequence = - p (Hence Proved).