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

Answer 1

(c) 26 is the correct choice since the total number of terms are 52 of which 26 terms get cancelled.

Answer 2

The correct option is (C) 26

Explanation:

We have to expand (x + a)⁵¹ – (x – a)⁵¹
At first,  (x + a)⁵¹ = ⁵¹C₀ x⁵¹ + ⁵¹C₁ x^50 . a + ⁵¹C₂ x^49 . a² + ...... + ⁵¹C₅₁ a⁵¹
then, (x – a)⁵¹ = ⁵¹C₀ x⁵¹ - ⁵¹C₁ x^50 . a + ⁵¹C₂ x^49 . a² - ...... - ⁵¹C₅₁ a⁵¹

When we subtract both the values i.e. (x + a)⁵¹ – (x – a)⁵¹ we get, 
2( ⁵¹C₁ x^50 . a + ⁵¹C₃ x^48 . a³ + ...... + ⁵¹C₅₁ a⁵¹)

Thus count the number of terms that is number of odd numbers up to 51. 

i.e 1, 3, 5, 7, ......, 49, 51

Apply AP:
a + (n-1)d = 51
1 + (n-1)2 = 51
n = 26
So, the total numbers of terms in the expansion is 26.