Discrete Maths - Mathematical Induction & Binomial Theorem

Explore binomial coefficients in the binomial expansion, showing that the sum of coefficients equals 2^n and that the sums of even and odd coefficients are equal.

Learn to find coefficients in binomial expansions using the general term T_{r+1}=nCr a^{n-r} b^r. Determine the coefficient of x in (x+1)^3 and apply r to get the desired term.

Prove the binomial theorem for positive integral index using mathematical induction, establishing the base case and the inductive step with binomial coefficients.

Learn how to reduce the binomial's first term to unity by using the one plus x form of the binomial theorem, with practical example calculations.

Explore the binomial theorem and binomial expressions, from simple cases to higher powers, using Pascal's triangle to identify coefficients and connections to combinations.

Derive the general term to expand powers of binomials using binomial coefficients, showing the pattern of terms C(n,r) a^(n-r) b^r and verifying with (x+y)^n.

Learn to find the middle term(s) in binomial expansions using the binomial theorem. Even n gives a single middle term, while odd n gives two, with examples.

Discover a new form of the binomial theorem for expansion, pattern, and application using one plus x to the n, with comparisons to the original form.

Explore the binomial theorem in a new form by expanding (1+x)^(1/2) and (1-x)^(1/2), revealing fractional-power series and coefficients like 1/2 and -1/8.

Introduce the generalized binomial expansion for non-positive exponents, deriving (1+x)^-1 as 1 - x + x^2 - x^3 + …, and apply it to compute reciprocals like 1/1.1.

Applying the binomial theorem to approximate sqrt(99) by expressing it as (100 − 1)^(1/2) and expanding around 100, achieving four-decimal accuracy.

Demonstrate a mathematical induction proof that 3^n minus 1 is divisible by seven for all natural numbers, including base case, inductive assumption, and the induction step.

expand the binomial (2x^2+3)^4 using the binomial theorem, derive terms for k=0 to 4, and obtain the expanded form 16x^8 + 96x^6 + 216x^4 + 216x^2 + 81.

Use binomial theorem to expand (1.1)^5 by computing binomial coefficients (5C0 to 5C5) and summing terms to obtain 1.61051.

Use the binomial theorem to expand ten minus point one cubed, collect coefficients, and obtain 970.29299 for 9.9 cubed.

Apply the binomial theorem to (x^2 - 4/x)^{11} to find the fifth term. Use T_{r+1}=C(11,r)a^{11-r}b^{r} with a=x^2 and b=-4/x to get 84480 x^{10}.

Compute the coefficient of x^9 in the expansion of (1/x + x^2)^18 using the binomial theorem; set -18+3r=9 to obtain r=9, giving 18C9=48620.

Identify the term independent of x in the binomial expansion with n=9, using binomial coefficients and r, including 9C3, to isolate the constant term.

Derive the first four terms of the given binomial expansion using the binomial theorem. Demonstrate the coefficients and powers, including 1, x/3, 2/9 x^2, and 14/27 x^3.

Demonstrate summing binomial coefficients c0 through c9 using the binomial theorem and verify the sum equals 2^9, highlighting the double formula.

Explore level-2 numericals on mathematical induction and binomial theorem through solved questions, covering induction proofs, the binomial middle terms, and practical expansion techniques.

Explore level-2 numericals on mathematical induction and the binomial theorem, including expansion comparisons, sum of coefficients, and coefficient extraction.