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If T_0, T_1, T_2, .....T_n represent the term in the expansion

Question

If $T_{0}, T_{1}, T_{2}, ..... T_{n}$ represent the term in the expansion of $(x + a)^{n}$ , then the value of $(T_{0} - T_{2} + T_{4} - T_{6} + .....)^{2} + (T_{1} - T_{3} + T_{5} + .....)^{2}$ is

(x^{2} - a^{2})^{n}

(x^{2} + a^{2})^{n}

(a^{2} - x^{2})^{n}

none of these

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Amelia Discussed

T_{0}, T_{1}, T_{2}, ..... T_{n}

represent the term in the expansion of

(x + a)^{n}

, then the value of

(T_{0} - T_{2} + T_{4} - T_{6} + .....)^{2} + (T_{1} - T_{3} + T_{5} + .....)^{2}

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Text solutionVerified

Given

T_{0}, T_{1}, T_{2}, ..............., T_{n}

represent terms in expansion of

(x + a)^{n}

(x + a)^{n} = T_{0} + T_{1} + T_{2} + T_{3} + ......... + T_{n} - - - - - - - - - (1)

When

a = - a

we get

(x - a)^{n} = T_{0} - T_{1} + T_{2} - T_{3} + ......... + (- 1)^{n} T_{n} - - - - - - (2)

Multiplying eq (1) and (2) and comparing with terms

x^{r}

we get

(x + a)^{n} \cdot (x - a)^{n} = (T_{0} - T_{2} + T_{4} - T_{6} + .........)^{2} + (T_{1} - T_{3} + T_{5} + .........)^{2}

(x^{2} - a^{2})^{n} = (T_{0} - T_{2} + T_{4} - T_{6} + .........)^{2} + (T_{1} - T_{3} + T_{5} + .........)^{2}

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Question Text	If $T_{0}, T_{1}, T_{2}, ..... T_{n}$ represent the term in the expansion of $(x + a)^{n}$ , then the value of $(T_{0} - T_{2} + T_{4} - T_{6} + .....)^{2} + (T_{1} - T_{3} + T_{5} + .....)^{2}$ is
Answer Type	Text solution:1
Upvotes	150