The total number of terms in the expansion of 1-x2 n-1-x2 n

In #160;(1+x)2n (1 x)2nSince(1+x)2n= #160;2nC0+ #160;2nC1x+.....+ #160;2nC2nx2n(1 x)2n= #160;2nC0 #160;2nC1x+......+ #160;2nC2nx2n #160; Subtra ...

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Standard XII

Mathematics

General Term of Binomial Expansion

The total num...

Question

The total number of terms in the expansion of (1 + x)²ⁿ – (1 – x)²ⁿ ___________.

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Solution

$In {(1 + x)}^{2 n} - {(1 - x)}^{2 n} Since {(1 + x)}^{2 n} =^{2 n} C_{0} +^{2 n} C_{1} x + . . . . . +^{2 n} C_{2 n} x^{2 n} {(1 - x)}^{2 n} =^{2 n} C_{0} -^{2 n} C_{1} x + . . . . . . +^{2 n} C_{2 n} x^{2 n}$
Subtracting above two,
$i . e . {(1 + x)}^{2 n} - {(1 - x)}^{2 n} = 2 [^{2 n} C_{1} x +^{2 n} C_{3} x^{3} + . . . .^{2 n} C_{2 n - 1} x^{2 n - 1}]$
$\begin{array}{rcl} i . e . number of terms here & = & 2 \times \frac{n}{2} (odd from 2 n is n) \\ = & n \end{array}$

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The total number of terms in the expansion of (1 + x)2n – (1 – x)2n ___________.

In (1+x)2n-(1-x)2nSince(1+x)2n= 2nC0+ 2nC1x+.....+ 2nC2nx2n(1-x)2n= 2nC0- 2nC1x+......+ 2nC2nx2n Subtracting above two, i.e. (1+x)2n-(1-x)2n=2 2nC1x+ 2nC3x3+.... 2nC2n-1x2n-1 i.e. number of terms here=2×n2 (odd from 2n is n)=n

The total number of terms in the expansion of (1 + x)²ⁿ – (1 – x)²ⁿ ___________.