The number of terms in the expansion of (√5+√[4]11)124 which are integers, is equal to

Question

The number of terms in the expansion of (√5+√[4]11)124 which are integers, is equal to (A) 0 (B) 30 (C) 31 (D) 32

Tardigrade Admin · Accepted Answer

The general term in the expansion of (√5+4 √11)124 is Tr+1= 124 Cr(√5)124-r(√[4]11)r = 124 Cr ⋅ 5(124-r) / 2 ⋅ 11r / 4 For integer terms r=0,4,8,12, ldots, 124 ∴ Number of terms which are integers =32

Q. The number of terms in the expansion of $(5 + 411)^{124}$ which are integers, is equal to

0

30

31

32

The general term in the expansion of $(5 + 411)^{124}$ is
$T_{r + 1} =^{124} C_{r} (5)^{124 - r} (411)^{r}$
$=^{124} C_{r} \cdot 5^{(124 - r) /2} \cdot 1 1^{r /4}$
For integer terms $r = 0, 4, 8, 12, \dots, 124$
$∴$ Number of terms which are integers $= 32$

Q. The number of terms in the expansion of (5​+411​)124 which are integers, is equal to

0

30

31

32

The general term in the expansion of (5​+411​)124 is Tr+1​=124Cr​(5​)124−r(411​)r =124Cr​⋅5(124−r)/2⋅11r/4 For integer terms r=0,4,8,12,…,124 ∴ Number of terms which are integers =32

Q. The number of terms in the expansion of $(5 + 411)^{124}$ which are integers, is equal to

The general term in the expansion of $(5 + 411)^{124}$ is
$T_{r + 1} =^{124} C_{r} (5)^{124 - r} (411)^{r}$
$=^{124} C_{r} \cdot 5^{(124 - r) /2} \cdot 1 1^{r /4}$
For integer terms $r = 0, 4, 8, 12, \dots, 124$
$∴$ Number of terms which are integers $= 32$