The number of integral terms in the expansion of (√3+√[8]5)256 is

Question

The number of integral terms in the expansion of (√3+√[8]5)256 is (A) 32 (B) 33 (C) 34 (D) 35

Tardigrade Admin · Accepted Answer

Tr+1= 256 Cr(√3)256-r 5(r/8) For integral terms (256-r/2), (r/8) are both positive integer ∴ r=0,8,16, ldots 256 ∴ 256=0+(n-1) 8 using tn=a+(n-1) d ∴ (256/8)=n-1 ∴ n=(256/8)+1 n =32+1=> n =33

Q. The number of integral terms in the expansion of $(3 + 85)^{256}$ is

$32$

$33$

$34$

$35$

$T_{r + 1} =^{256} C_{r} (3)^{256 - r} 5^{\frac{r}{8}}$
For integral terms $\frac{256 - r}{2}, \frac{r}{8}$ are both positive
integer
$∴ r = 0, 8, 16, \dots 256$
$∴ 256 = 0 + (n - 1) 8$ using $t_{n} = a + (n - 1) d$
$∴ \frac{256}{8} = n - 1$
$∴ n = \frac{256}{8} + 1$
$n = 32 + 1 => n = 33$

Q. The number of integral terms in the expansion of (3​+85​)256 is

32

33

34

35