Question

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

A
128
B
129
C
130
D
131

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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