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I am trying to answer the following problem but I barely understand the question and I've no idea how to proceed.

Calculate the value of $\sum\limits_{k = 0}^n 5^k\binom nk$ for cases where $n = 1,\;2,\;3$.

Could somebody please give me some pointers?

Thanks in advance

Rushabh Mehta
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Emilycodes
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  • You'll want to recall the binomial expansion formula:

    $$(a+b)^n = \sum_{k=0}^n \binom n k a^k b^{n-k}$$

    Consider values of $a,b$ that help you match the formula you wish to calculate. For example, letting $b=1$ (and thus getting rid of the $b^{n-k}$ term) is quite tempting...

     – PrincessEev Dec 07 '19 at 06:13
  • Thank you for this. – Emilycodes Dec 07 '19 at 06:24

1 Answers1

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By the Binomial Theorem, we have $$\sum_{k=0}^n5^k\binom nk=\sum_{k=0}^n5^k1^{n-k}\binom nk=(5+1)^n=6^n.$$ So, plugging $n=1,2,3$, we get $\boxed{6,36,216}$.

ViHdzP
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