The total number of terms in the expansion of (x + a)51 – (x – a)5

1.	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» (c) 26 is the correct choice since the total number of terms are 52 of which 26 terms get cancelled. 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 x⁴⁹^.a² + ...... + ⁵¹C₅₁a⁵¹ then, (x – a)⁵¹ = ⁵¹C₀ x⁵¹ - ⁵¹C₁ x^50 .a + ⁵¹C₂ x⁴⁹^.a² - ...... - ⁵¹C₅₁a⁵¹ 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 + ...... + ⁵¹C₅₁a⁵¹) 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.

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»

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 x⁴⁹^.a² + ...... + ⁵¹C₅₁a⁵¹
then, (x – a)⁵¹ = ⁵¹C₀ x⁵¹ - ⁵¹C₁ x^50 .a + ⁵¹C₂ x⁴⁹^.a² - ...... - ⁵¹C₅₁a⁵¹

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 + ...... + ⁵¹C₅₁a⁵¹)

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.

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