Find the term indicated in each expansion. (x2 + y3)8; sixth term

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problem 43

Chapter 9, Problem 43

Find the term indicated in each expansion. (x²+ y³)⁸; sixth term

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Identify the general term in the binomial expansion of $\left(x^{2} + y^{3}\right)^8$. The general term is given by the formula: $T_{k+1} = \binom{8}{k} (x^{2})^{8-k} (y^{3})^{k}$, where $k$ ranges from 0 to 8.

Since we want the sixth term, set $k + 1 = 6$, which means $k = 5$.

Substitute $k = 5$ into the general term formula: $T_6 = \binom{8}{5} (x^{2})^{8-5} (y^{3})^{5}$.

Simplify the exponents: $(x^{2})^{3} = x^{6}$ and $(y^{3})^{5} = y^{15}$, so the term becomes $\binom{8}{5} x^{6} y^{15}$.

Calculate the binomial coefficient $\binom{8}{5}$ using the formula $\binom{n}{r} = \frac{n!}{r!(n-r)!}$, then write the full expression for the sixth term.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n into a sum of terms involving binomial coefficients. Each term is given by C(n, k) * a^(n-k) * b^k, where C(n, k) is a combination representing the number of ways to choose k elements from n.

Binomial Coefficients and Combinations

Binomial coefficients, denoted as C(n, k) or "n choose k," represent the number of ways to select k items from n without regard to order. They are calculated using factorials: C(n, k) = n! / (k! * (n-k)!). These coefficients determine the coefficients of terms in a binomial expansion.

Term Identification in Binomial Expansion

In the expansion of (a + b)^n, the (k+1)-th term corresponds to C(n, k) * a^(n-k) * b^k. To find a specific term, identify k based on the term number, then substitute into the formula. For example, the sixth term corresponds to k = 5.

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