Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • The number of integral terms in the expansion of <img alt="\small \left(5^{1 / 2}+7^{1 / 8}\right)^{1024}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Csmall%20%5Cleft%285%5E%7B1%20/%202

The number of integral terms in the expansion of \small \left(5^{1 / 2}+7^{1 / 8}\right)^{1024} is 
 

Option: 1

128


Option: 2

129


Option: 3

130


Option: 4

131


Answers (1)

best_answer

The general term in the expansion of  \left(5^{1 / 2}+7^{1 / 8}\right)^{1024} is \begin{aligned} & \mathrm{T}_{\mathrm{r}+1}={ }^{1024} C_r\left(5^{1 / 2}\right)^{1024-r}\left(7^{1 / 8}\right)^r \\ & ={ }^{1024} C_r 5^{512-r / 2} 7^{r / 8}=\left({ }^{1024} C_r 5^{512-r}\right)\left(5^{r / 2} 7^{r / 8}\right) \end{aligned}

=\left\{{ }^{1024} C, 5^{512-r}\right\}\left\{5^4 \times 7\right\}^{r / 8}

Clearly, T_{r+1} will be an integer, iff r/8 is an integer such that 0 ≤ r ≤ 1024. 
    ⇒  r is a multiple of 8 lying satisfying 0 ≤ r ≤ 1024 
    ⇒     r = 0, 8, 16, . . . , 1024 
    ⇒     r can assume 129 values 
    Hence, there are 129 integral terms in the expansions of  \left(5^{1 / 2}+7^{1 / 8}\right)^{1024}.
 

Posted by

Riya

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JoSAA 2025 reconstituted; NIT Rourkela to coordinate BTech counselling for IITs, NITs, IIITs
anam-khan

JEE Main 2025: BTech Information Technology cut-offs at NITs; know opening, closing ranks
anam-khan

JEE Main 2025: BTech Computer Science Engineering cut-offs at NITs went up this year

Latest Question