The total number of terms in the expansion of

Question:

The total number of terms in the expansion of (x + a)51 – (x – a)51 after simplification is

(a) 102

(b) 25

(c) 26

(d) none of these

Solution:

for $(x+a)^{51}-(x-a)^{51}$

Since $(x+a)^{51}={ }^{51} C_{0} x^{51}+{ }^{51} C_{1} x^{50} a+{ }^{51} C_{2} x^{49} a^{2}+\ldots+{ }^{51} C_{51} a^{51}$

and $(x-a)^{51}={ }^{51} C_{0} x^{51}-{ }^{51} C_{1} x^{50} a+{ }^{51} C_{2} x^{49} a^{2} \ldots{ }^{51} C_{51} a^{51}$

Subtracting above values,

$(x+a)^{51}-(x-a)^{51}=2\left({ }^{51} C_{1} x^{50} a+{ }^{51} C_{3} x^{48} a^{3}+\ldots+{ }^{51} C_{51} a^{51}\right)$

i.e.      1st, 3rd, 5th, 7th ________  49th, 51th term are there

∴  applying A.P.           

a + ( n – 1)= 51

i.e. 1 + (n – 1) 2 = 51

i.e. 2(n – 1) = 50

i.e. = 26

Hence, the correct answer is option C.

 

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