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Question

In the expansion of $$\left (5^{ \tfrac {1}{2}}+7^{\tfrac {1}{8}}\right )^{1024}$$, the number of integral terms is


A
128
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B
129
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C
130
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D
131
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Solution

The correct option is B $$129$$
The general term in the expansion of $$(5^{\tfrac 12} + 7^{\tfrac 18})^{1024}$$ is
$$T_{r+1} = ^{1024}C_r (5^{\tfrac 12})^{1024-r} (7^{\tfrac 18})^r$$
$$= ^{1024}C_r 5^{512-\tfrac{r}{2}} 7^{\tfrac{r}{8}}$$
$$ = (^{1024}C_r 5^{512-r})(5^{\tfrac{r}{2}}7^{\tfrac{r}{8} })$$
$$= (^{1024}C_r 5^{512r}) (5^4\times 7)^{\tfrac{r}{8}}$$
Clearly $$T_{r+1}$$ will be an integer, if $$\dfrac{r}{8}$$ is an integer such that $$0\leq r\leq 1024$$
$$\Rightarrow r$$ is a multiple of $$8$$ lying satisfying $$0\leq r \leq 1024$$
$$ r = 0, 8, 16, ... 1024$$
$$\Rightarrow r$$ can assume $$129$$ values.
Hence, there are $$129$$ integral terms in the expansion of $$(5^{\tfrac 12} + 7^{\tfrac 18})^{1024}$$

Mathematics

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