The Binomial Theorem Solution

                                     

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The Binomial Theorem Solution

The Binomial Theorem is a powerful algebraic tool that allows us to expand expressions of the form (a + b)^n, where a and b are real numbers, and n is a positive integer.

To prove the Binomial Theorem, we first need to establish the following lemma:

Lemma: For any positive integer k, we have (1 + x)^k = 1 + kx + (k choose 2)x^2 + ... + (k choose k)x^k, where (k choose i) denotes the binomial coefficient.

Proof of lemma: We can prove the lemma by using mathematical induction on k. The base case is k = 1, where we have (1 + x)^1 = 1 + x, which is true.

Assume that the lemma is true for some positive integer k. We will show that it is also true for k+1.

We have (1 + x)^(k+1) = (1 + x)^k * (1 + x). Using the induction hypothesis, we can expand (1 + x)^k as

The Binomial Theorem is a powerful algebraic tool that allows us to expand expressions of the form (a + b)^n, where a and b are real numbers, and n is a positive integer.

To prove the Binomial Theorem, we first need to establish the following lemma:

Lemma: For any positive integer k, we have (1 + x)^k = 1 + kx + (k choose 2)x^2 + ... + (k choose k)x^k, where (k choose i) denotes the binomial coefficient.

Proof of lemma: We can prove the lemma by using mathematical induction on k. The base case is k = 1, where we have (1 + x)^1 = 1 + x, which is true.

Assume that the lemma is true for some positive integer k. We will show that it is also true for k+1.

We have (1 + x)^(k+1) = (1 + x)^k * (1 + x). Using the induction hypothesis, we can expand (1 + x)^k as

(1 + x)^k = 1 + kx + (k choose 2)x^2 + ... + (k choose k)x^k.

Multiplying this expression by (1 + x), we obtain

(1 + x)^{k+1} = (1 + kx + (k choose 2)x^2 + ... + (k choose k)x^k) * (1 + x)

Expanding this product using distributivity, we get

(1 + x)^{k+1} = 1 + (k+1)x + ((k choose 2) + k)x^2 + ... + ((k choose k) + 1)x^k + x^{k+1}

Note that we used the identity (k choose i) + (k choose i-1) = (k+1 choose i) to simplify the coefficients of the terms x^i for i = 2, 3, ..., k.

This completes the proof of the lemma.

Now, using the lemma, we can easily derive the Binomial Theorem. To expand (a + b)^n, we can substitute x = b/a in the expression (1 + x)^n and multiply both sides by a^n. This gives us

(a + b)^n = a^n + n a^{n-1} b + (n choose 2) a^{n-2} b^2 + ... + (n choose k) a^{n-k} b^k + ... + b^n

This is the desired formula for the Binomial Theorem.

Thus, we have proved the Binomial Theorem using a mathematical induction argument based on the lemma.