Question

If n is the number of irrational terms in the expansion of `(3^(1/4) + 5^(1/8))^60`, then (n – 1) is divisible by ______.

पर्याय

• 8

• 26

• 7

• 30

Answer 1

If n is the number of irrational terms in the expansion of `(3^(1/4) + 5^(1/8))^60`, then (n – 1) is divisible by 26.

Explanation:

Given binomial expression is `(3^(1/4) + 5^(1/8))^60`

For (A + B)ⁿ,

T_r+1 = ⁿC_r(A)^n-r(B)^r;0 ≤ r ≤ n;r ∈ W

Using the above concept, we can write

T_r+1 = `""^60C_r (3^(1/4))^(60-r)(5^(1/8))^r`

⇒ T_r+1 = `""^60C_r(3)^((60-r)/4)(5)^(r/8)`  ...(i)

As 0 ≤ r ≤ n ⇒ 0 ≤ r ≤ 60  ...(ii)

⇒ `0 ≤ r/8 ≤ 60/8`

⇒ `0 ≤ r/8 ≤ 7.5`

For rational terms

`r/8 ∈{0, 1, 2, 3, 4, 5, 6, 7}`  ...(iii)

Using equation (ii),

0 ≤ r ≤ 60

–60 ≤ –r ≤ 0

0 ≤ 60 – r ≤  60

`0 ≤ (60 - r)/4 ≤ 15`

For rational terms

`(60 - r)/4 ∈{0, 1, 2, 3, 4, ........., 15}`  ...(iv)

 Using Equations (iii) and (iv),

Total rational terms = 8

Total number of terms = 60 + 1 = 61

Hence total irritational number of terms = 61 – 8 = 53

Given, n = Number of irrational terms

n = 53

⇒ n – 1 = 53 – 1

⇒ n – 1 = 52

⇒ 52 is divisible by 26