The number of integral terms in the expansion | vedclass

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(B) The general term in the expansion of $(7^{1/2} + 11^{1/6})^{824}$ is given by $T_{r+1} = {}^{824}C_r (7)^{(824-r)/2} (11)^{r/6}$.For the term to b

Vedclass · Accepted Answer

$138$

Vedclass · Answer

(B) The general term in the expansion of $(7^{1/2} + 11^{1/6})^{824}$ is given by $T_{r+1} = {}^{824}C_r (7)^{(824-r)/2} (11)^{r/6}$.For the term to be an integer,the exponents of $7$ and $11$ must be integers.$1$. The exponent of $7$ is $(824-r)/2 = 412 - r/2$. For this to be an integer,$r$ must be an even number.$2$. The exponent of $11$ is $r/6$. For this to be an integer,$r$ must be a multiple of $6$.Combining these,$r$ must be a multiple of $	ext{lcm}(2, 6) = 6$.Thus,$r$ can take values $0, 6, 12, \dots, 822$.This is an arithmetic progression where $a = 0$,$d = 6$,and the last term $l = 822$.Using the formula $l = a + (n-1)d$,we get $822 = 0 + (n-1)6$.$n-1 = 822/6 = 137$.$n = 138$.Therefore,there are $138$ integral terms.

The number of integral terms in the expansion of $\{7^{(1/2)} + 11^{(1/6)}\}^{824}$ is equal to ...................

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