# Is this a correct way to rewrite the binomial theorem?

• jey1234
In summary, the conversation discusses using the binomial theorem in a proof and rewriting it using sigma notation. The correct way to extract the first term of the binomial expansion is to increase the lower bound by 1 and write the first term outside the summation. It is also confirmed that the given equation is correct.
jey1234

## Homework Statement

I am doing a poof and I need to use the binomial theorem. However is the following a correct way to rewrite it?

$$(a+b)^n\ =\ {n \choose 0}a^{n} + \sum_{k=1}^{n}{n \choose k}\ a^{n-k}\ b^{k}$$

## Homework Equations

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

## The Attempt at a Solution

Basically, I want to extract the first term of the binomial expansion out of the summation but I'm not that good with sigma notation. Don't I just have to increase the lower bound by 1 and write the first term outside (as shown above)? Thanks.

hey jey1234!

yes, that's fine

thanks tim. one more quick question. is the following correct?

$$\frac{1}{a}\sum_{k=1}^{n}a^{k}\ b^{k}\ =\ \sum_{k=1}^{n}a^{k-1}\ b^{k}$$

jey1234 said:
thanks tim. one more quick question. is the following correct?

$$\frac{1}{a}\sum_{k=1}^{n}a^{k}\ b^{k}\ =\ \sum_{k=1}^{n}a^{k-1}\ b^{k}$$
Yes, that's correct.

thanks sammy

## 1. What is the binomial theorem?

The binomial theorem is a mathematical formula used to expand the power of a binomial expression (an expression with two terms) raised to a positive integer exponent. The theorem states that the expansion will result in a sum of terms, each with a coefficient and a variable raised to a power.

## 2. Is there only one correct way to rewrite the binomial theorem?

No, there are multiple ways to rewrite the binomial theorem, but all of them follow the same basic structure and principles.

## 3. What are the key components of the binomial theorem?

The key components of the binomial theorem include the binomial coefficients, the variables, and the exponents. The binomial coefficients are the numbers in front of the variables and represent the number of ways that the terms can be combined. The variables are the letters or symbols used to represent unknown quantities, and the exponents are the powers to which the variables are raised.

## 4. Can the binomial theorem be used for any values of the variables and exponent?

Yes, the binomial theorem can be used for any values of the variables and exponent. However, it is typically used for positive integer exponents, as this is when the formula is most useful and efficient.

## 5. How is the binomial theorem used in real-world applications?

The binomial theorem has many practical applications in fields such as physics, chemistry, and engineering. It is used to expand and simplify mathematical expressions, make predictions in probability and statistics, and solve equations in various scientific and engineering problems.

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