Explanation:
A Given, \((x+a)^{47}-(x-a)^{47}\)
Then, expansion of \((x+a)^{47}\) -
\((x-a)^{47}=\)
\({ }^{47} C_0 x^{47} a^0+{ }^{47} C_1 x^{46} a^1+{ }^{47} C_2 x^{45} a^2+\ldots \ldots .+{ }^{47} C_{47} x^0 a^{47}\)
And, expansion of \((x-a)^{47}\) -
\((x-a)^{47}={ }^{47} \mathrm{C}_0 \mathrm{x}^{47} \mathrm{a}^0-{ }^{47} \mathrm{C}_1 \mathrm{x}^{46} \mathrm{a}^1-\ldots .+{ }^{47} \mathrm{C}_{47} \mathrm{x}^0 \mathrm{a}^{47}\)
On subtracting equation (ii) by equation (i), we get-
\((x+a)^{47}-(x-a)^{47}=2\left[{ }^{47} \mathrm{C}_1 \mathrm{x}^{46} \mathrm{a}^1+{ }^{47} \mathrm{C}_3 \mathrm{x}^{43} \mathrm{a}^3+\ldots . .+{ }^{47} \mathrm{C}_{47} \mathrm{x}^0 \mathrm{a}^{47}\right]\)
We see that, odd term \(1,3,5, \ldots \ldots, 47\) are in A. P.
Then,
\(47=1+(n-1) \times 2\)
\(47=1+2 n-2\)
\(47+2=1+2 n\)
\(49=1+2 n\)
\(2 n=48\)
\(n=24\)So, total number of terms is 24 .
