Question
The number of terms in the expansion of $${\left( {{x^2} + 1 + \frac{1}{{{x^2}}}} \right)^n},n \in N,$$ is
A.
$$2n$$
B.
$$3n$$
C.
$$2n + 1$$
D.
$$3n + 1$$
Answer :
$$2n + 1$$
Solution :
$${\left\{ {1 + \left( {{x^2} + \frac{1}{{{x^2}}}} \right)} \right\}^n} = {\,^n}{C_0} + {\,^n}{C_1}\left( {{x^2} + \frac{1}{{{x^2}}}} \right) + {\,^n}{C_2}{\left( {{x^2} + \frac{1}{{{x^2}}}} \right)^2} + ..... + {\,^n}{C_n}{\left( {{x^2} + \frac{1}{{{x^2}}}} \right)^n}$$
Here, all the terms are positive and will contain powers
$${\left( {{x^2}} \right)^0},{\left( {{x^2}} \right)^1},{\left( {{x^2}} \right)^2},.....,{\left( {{x^2}} \right)^n},{\left( {{x^2}} \right)^{ - n}},{\left( {{x^2}} \right)^{ - \left( {n - 1} \right)}},.....,{\left( {{x^2}} \right)^{ - 1}}\,{\text{only}}{\text{.}}$$
∴ the number of terms will be $$2n + 1.$$