The number of irrational terms in the expansi | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let $E = (5^{1/2} + 7^{1/8})^{1024} + (5^{1/2} - 7^{1/8})^{1024}$.Using the binomial expansion,the general term $T_{r+1}$ of $(a+b)^n + (a-b)^n$ i

Vedclass · Accepted Answer

$384$

Vedclass · Answer

(D) Let $E = (5^{1/2} + 7^{1/8})^{1024} + (5^{1/2} - 7^{1/8})^{1024}$.Using the binomial expansion,the general term $T_{r+1}$ of $(a+b)^n + (a-b)^n$ is $2 	imes \sum_{k=0, 2, 4, \dots} \binom{n}{k} a^{n-k} b^k$.Here $n = 1024$,$a = 5^{1/2}$,and $b = 7^{1/8}$.The terms are of the form $2 \binom{1024}{k} (5^{1/2})^{1024-k} (7^{1/8})^k = 2 \binom{1024}{k} 5^{512 - k/2} 7^{k/8}$.For the term to be rational,$k/2$ and $k/8$ must be integers,which implies $k$ must be a multiple of $8$.Since $0 \le k \le 1024$ and $k$ is even,$k \in \{0, 8, 16, \dots, 1024\}$.The number of such values of $k$ is $\frac{1024}{8} + 1 = 128 + 1 = 129$.These $129$ terms are rational.The total number of terms in the expansion of $(5^{1/2} + 7^{1/8})^{1024}$ is $1024 + 1 = 1025$.However,in the sum $(5^{1/2} + 7^{1/8})^{1024} + (5^{1/2} - 7^{1/8})^{1024}$,the odd terms cancel out,leaving only the even terms.The number of terms in the resulting expansion is $\frac{1024}{2} + 1 = 513$.Out of these $513$ terms,$129$ are rational.Therefore,the number of irrational terms is $513 - 129 = 384$.

The number of irrational terms in the expansion of $(5^{1/2} + 7^{1/8})^{1024} + (5^{1/2} - 7^{1/8})^{1024}$ is

Similar Questions

If $(1+x)^{15}=a_0+a_1 x+\ldots+a_{15} x^{15}$,then $\sum_{r=1}^{15} r \frac{a_r}{a_{r-1}}$ is equal to

The coefficient of $x^{24}$ in the expansion of $(1+x^2)^{12}(1+x^{12})(1+x^{24})$ is

In the expansion of $(1+x)^n$,the coefficients of the $p^{th}$ and $(p+1)^{th}$ terms are respectively $p$ and $q$. Then $p+q$ is equal to:

Find the coefficient of $x^{64}$ in the expansion of $P(x) = (x - 1)^2(x - 2)^3(x - 3)^4 \dots (x - 10)^{11}$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module