Question

the m'th term of an AP is n and n'th term is m.1st term is equal to what????arithmetic mean of t23 and t25????​

Answer 1

Answer:

Given:

T(m) = n ---------(1)

T(n) = m --------(2)

Let, the first term of the AP be a and and common difference be d.

Thus, From eq-(1) and (2) , we have;

=> T(m) = n

=> a + (m - 1)d = n ---------(3)

Also,

=> T(n) = m

=> a + (n - 1)d = m --------(4)

Now, subtracting eq-(4) from (3) ,

We get;

=> {a + (m - 1)d} - {a + (n - 1)d} = n - m

=> a + (m - 1)d - a - (n - 1)d = n - m

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

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

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

=> d = - 1

Now,

Putting the value of d = -1 in eq-(3),

We get;

=> a + (m - 1)d = n

=> a + (m - 1)(-1) = n

=> a - m + 1 = n

=> a = m + n - 1

Hence, the first term of the AP must be:

m + n - 1.

Note: The arithmetic mean of two numbers P and Q is given by ;

AM = (P + Q)/2

Now;

=> T(23) = a + (23 - 1)d

= a + 22d

= m + n - 1 + 22(-1)

= m + n - 1 - 22

= m + n - 23

Also;

=> T(25) = a + (25 - 1)d

= a + 24d

= m + n - 1 + 24(-1)

= m + n - 1 - 24

= m + n - 25

Now;

The arithmetic mean of T(23) and T(23) will be given by;

AM = { T(23) + T(25) }/2

= { m + n - 23 + m + n - 25 }/2

= { 2m + 2n - 48 }/2

= 2m/2 + 2n/2 - 48/2

= m + n - 24

Hence, the arithmetic mean of 23rd term and 25th term of the AP is:

m + n - 24.