Question

Q6. The Binomial Theorem says that given any $a, b \in \mathbb{R}$ and an integer $n \ge 0$ $(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$ Prove the formula and use it to prove the identities: 1. $\sum_{k=0}^n \binom{n}{k} = 2^n$ 2. $\sum_{k=0}^n \binom{n}{k} 9^k = 10^n$ What is the coefficient of $x^7$ in $(2x + 3)^{10}$ when it is expanded by the binomial theorem?

          Q6. The Binomial Theorem says that given any $a, b \in \mathbb{R}$ and an integer $n \ge 0$

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

Prove the formula and use it to prove the identities:

1. $\sum_{k=0}^n \binom{n}{k} = 2^n$
2. $\sum_{k=0}^n \binom{n}{k} 9^k = 10^n$

What is the coefficient of $x^7$ in $(2x + 3)^{10}$ when it is expanded by the binomial theorem?

Q6. The Binomial Theorem says that given any a, b ∈ℝ and an integer n ≥ 0

(a + b)^n = ∑k=0^n nk a^n-k b^k

Prove the formula and use it to prove the identities:

1. ∑k=0^n nk = 2^n
2. ∑k=0^n nk 9^k = 10^n

What is the coefficient of x^7 in (2x + 3)^10 when it is expanded by the binomial theorem?

Added by Stephen F.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert James Kiss

Step 1

To prove the Binomial Theorem, we can use mathematical induction. Show more…

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Q6. The Binomial Theorem says that given any a, b âˆˆ R and an integer n > 0, the formula is (a + b)^n = Î£(k=0 to n) (nCk) * a^(n-k) * b^k. Prove the formula and use it to prove the identities: 1. Î£(k=0 to n) (nCk) = 2^n 2. Î£(k=0 to n) (nCk) * k = n * 2^(n-1) What is the coefficient of x^7 in (2x + 3)^10 when it is expanded by the binomial theorem?