Question

number of rational terms in the expansion (√2 + 3^1/4)^100 is:

Answer 1

Answer:

26

Step-by-step explanation:

Hi,

Given expression is (√2 + 3¹/⁴ )¹⁰⁰

General term of the above expansion is given by

Tₓ = ¹⁰⁰Cₓ(√2)ˣ(3¹/⁴)¹°°⁻ˣ

Tₓ = ¹⁰⁰Cₓ(2)ˣ/²(3)²⁵⁻ˣ/⁴

For the terms to be rational,

the terms in the general term should be free from radicals

Power of 2 is x/2 which should be an integer

⇒ x should be multilpe of 2---------------(1)

Similarly, Power of 3 should be integer

Hence 25 - x/4 should be integer

⇒x/4 should be integer

x should be multiple of 4-------------(2)

From (1) and (2), we can conclude that x should be

multiple of 4.

Since x ranges from 0 to 100,

there are 26 value of x starting from 0, 4 ....

ending at 100 which are multiple of 4.

Hence, there are 26 terms  in total which are

rational.

Hope, it helps !