If the sum of odd numbered terms and the sum of even numbered ter… - Vidyadip

MCQ

If the sum of odd numbered terms and the sum of even numbered terms in the expansion of $(\text{x}+\text{a})^{\text{n}}$ are A and B respectively, then the value of $(\text{x}^{2}-\text{a}^{2})^{\text{n}}$ is:

✓
$A ^2- B ^2$
B
$A^2+B^2$
C
4 AB
D
None of these.

✓

Answer

Correct option: A.

$A ^2- B ^2$

$A ^2- B ^2$

Solution:
If A and B denote respectively the sums of odd terms and even terms in the expansion $(\text{x}+\text{a})^{\text{n}}.$
Then,
$(\text{x}+\text{a})^{\text{n}}=\text{A}+\text{B}\ ...(\text{i})$
$(\text{x}-\text{a})^{\text{n}}=\text{A}-\text{B}\ ...(\text{ii})$
Multplying both the equations we get,
$(\text{x}+\text{a})^{\text{n}}(\text{x}-\text{a})^{\text{n}}=\text{A}^{2}-\text{B}^{2}$
$\Rightarrow (\text{x}-\text{a})^{\text{n}}=\text{A}^{2}-\text{B}^{2}$

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