1
$\begingroup$

Textbook Question:

Given $\sum\limits_{i=0}^\mathbb{50}{50\choose i}8^i $ = $x^{100}$ . Find the values of x ?

Textbook Solution:

$\sum\limits_{i=0}^\mathbb{50}{50\choose i}8^i $ = $(1+8)^{50} = 9^{50} = [(+-3)^2)]^{50}$

$\therefore$ x = +-3

Personal Logical Question:

  1. In the given textbook solution, how is $(1+8)^{50}$ obtained?
  • 1
    Binomial Theorem. – 2017-02-20

2 Answers 2

2

The book is using Binomial Theorem:

$$\sum_{i=0}^{n}{n \choose i}a^ib^{n-i}=(a+b)^{n}$$

Use $a=8$, $b=1$ and $n=50$.

  • 0
    Ok, this makes sense, so a = 8 comes from $8^i$, but b = 1 is not mentioned here because anything to the power of 1 is 1 itself. I believe that was the missing information, am I right ? – 2017-02-20
  • 1
    @Kourosh: There is no missing information once $8^{i}=8^{i}\cdot 1^{50-i}$. That is a very common approach for that kind of question. – 2017-02-20
  • 0
    I see, yes as I thought before "anything to the power of 1 is 1 itself" , so it is not explicitly mentioned for $8^i$ , but as you suggested, it is assumed to exist. @Arnaldo, thank you for your help. Great clarification. – 2017-02-20
  • 0
    @Kourosh: you are very welcome! – 2017-02-20
0

The result you ask about, as Display name notes, is called the Binomial Theorem, and is a standard result. The Wikipedia page for it, unlike most, is actually quite useful.