The number of integral terms in the expansion of (5^(1/2) + 7^(1/8) )^(1024) is


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Question Number 86496 by ram roop sharma last updated on 29/Mar/20

$The number of integral terms in the$ $expansion of {(5^{\frac{1}{2}} + 7^{\frac{1}{8}})}^{1024} is$
Commented by Serlea last updated on 29/Mar/20

$1025$
Commented by mr W last updated on 29/Mar/20

$\underset{k = 0}{\sum^{1024}} C_{k}^{1024} 5^{\frac{k}{2}} 7^{\frac{1024 - k}{8}}$ $a t e r m i s i n t e g r a l w h e n$ $k = 8 r$ $\frac{1024 - 8 r}{8} = 128 - r ⩾ 0$ $\Rightarrow 0 ⩽ r ⩽ 128$ $\Rightarrow t h e r e a r e 129 i n t e g r a l t e r m s,$ $t h e s u m o f t h e m i s \underset{r = 0}{\sum^{128}} C_{8 r}^{1024} 5^{4 r} 7^{128 - r}$
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