Find the degree of
the expression _{.}

Since _{}

= _{}

=_{}

\ Degree equal to 7.

Show that there will be no
term containing x^{2r} in the expansion of (x + x^{-2})^{n-3}
if n –2r is positive but not a multiple of 3.

If possible suppose that the k^{th}
term contains x^{2r} in the expansion of

(x + x^{-2})^{n-3} , k^{th} term is given by

=

since the k^{th}
term contains x^{2r} \ 2r = n –3k ….(1).

or 3k = n –2r …..(2)

Now if n –2r is positive and not divisible by 3 then it is obvious from eqn. (2) that k will not be a positive integer.

Hence if (n –2r) is not
divisible by 3 there will be no term containing x^{2r} in the given
expansion. .

If x > 0 and the 4^{th}
term in the expansion of has maximum
value then x > 64/21.

Since T_{4} is the greatest term in
the expansion of ,

T_{3} < T_{4}
and T_{5} < T_{4} <
1 and > 1

But

=

\ 1 > = .

Also 1 < = .

Thus x > 64/21.

The coefficient of a^{10}b^{7}c^{3}
in the expansion of ( bc + ca + ab)^{10} is

The general term in the
expansion of ( bc + ca + ab)^{10} is

where r + s + t = 10

For the coefficient of a^{10}b^{7}c^{3
}we set t + s = 10, r + t = 7, r + s = 3.

Since r + s + t = 10 we get r = 0, s = 3, t = 7. .

Thus, the coefficient of a^{10}b^{7}c^{3}
in the expansion of ( bc + ca + ab)^{10} is

.

Show that 3^{2n+2}
– 8n – 9 is divisible by 64, " n∈N.

3^{2n+2} – 8n–9 = 3^{2}(3^{2})^{n}
–8n–9 = 9(1+8)^{n}–8n – 9

= 9 _{}

= 64 n +9 (^{n}C_{2}8^{2}
+ ^{n}C_{3}.8^{3} + -------- +8^{n}).

= 64 [n+9(^{n}C_{2}+^{n}C_{3}.8------+8^{n-2})].

= 64K where K is an integer.

Hence 3^{2n+2} – 8n–9 is divisible
by 64.