Problem 37 - Algebra and Trigonometry

Question

Find the 100 th term in the expansion of \((1+y)^{100}\)

Step-by-Step Solution

Answer

The 100th term is \(100y^{99}\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that any positive integer power of a binomial is the sum of the terms of the form \( \binom{n}{k} x^{n-k} y^k \). In this case, \(n = 100\) for \((1+y)^{100}\).

Step 2: Identify the General Term in Binomial Expansion

The general term \( T_{k+1} \) in the expansion of \((1+y)^{100}\) can be written as \( T_{k+1} = \binom{100}{k} (1)^{100-k} y^k = \binom{100}{k} y^k \).

Step 3: Determine the 100th Term

Since the general term is \( T_{k+1} = \binom{100}{k} y^k \), we need to find \(k\) such that \(k + 1 = 100\). Therefore, \(k = 100 - 1 = 99\).

Step 4: Calculate the Binomial Coefficient

The binomial coefficient \( \binom{100}{99} \) can be simplified using the property \( \binom{n}{k} = \binom{n}{n-k} \). Thus, \( \binom{100}{99} = \binom{100}{1} = 100 \).

Step 5: Formulate the 100th Term

Substituting \(k = 99\) into the general term gives \( T_{100} = \binom{100}{99} y^{99} = 100y^{99} \).

Key Concepts

Binomial Expansion

The binomial expansion involves expressing a binomial raised to any power as a sum of terms. It is a way to expand expressions that have exponents, allowing us to simplify calculations and understand polynomial behavior. For any expression of the form

• a binomial,
• like a + b,
• when raised to a power,

say \(n\),we can use the **Binomial Theorem.**This theorem gives us a formula to expand \((a+b)^{n}\)into a series of terms

• where each term includes a binomial coefficient and powers of individual terms.
• In the expression \((1+y)^{100}\), 100 represents the power to which the binomial is raised.

The terms of this expansion decrease the powers of one part of the binomial while increasing the powers of the other part. This makes it useful in calculations involving large powers because you can focus on only the terms you need.

Binomial Coefficients

In the binomial expansion, binomial coefficients are key players. They are the numbers that help us determine the value of each term in the expansion. Mathematically,

• these are denoted by \(\binom{n}{k}\),
• which is read as "n choose k,"

and they determine how many ways you can choose \(k\) elements from a set of \(n\)elements. **How to Calculate: **The formula for determining a binomial coefficient is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where!stands for factorial,which means multiplying a series of descending natural numbers. For example:

• 5! = 5×4×3×2×1 = 120.
• To find the 100th term in \((1+y)^{100}\),
• we calculated \(\binom{100}{99}\)
• which simplifies to 100 because\(\binom{n}{n-1} = n\).

Understanding and calculating these coefficients efficiently is imperative as they tell us the weight of each term.

Algebraic Expressions

Algebraic expressions are a combination of variables and constants using the operations of addition, subtraction, multiplication, division, and exponentiation. The binomial formula constructs an algebraic expression as a sum of terms, each having:

• a fixed numerical factor, known as a binomial coefficient,
• and a variable factor made up of powers of terms from the binomial.

In context:

• For \((1+y)^{100}\),
• the expression is expanded into an algebraic form where each term consists of a binomial coefficient and the variable \(y\).

**Simplifying Algebraic Expressions:**Once expanded, you can simplify by adding or subtracting like terms, which are the terms with the same variables raised to the same powers. This particularly comes in handy when solving equations or making predictions using algebraic models. Understanding algebraic expressions through binomial expansion makes complex mathematical problems more manageable by breaking them into simpler parts.