Resources tagged with: Binomial Theorem

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Broad Topics > Algebraic expressions, equations and formulae > Binomial Theorem

The Kth Sum of N Numbers

Age 16 to 18

Yatir from Israel describes his method for summing a series of triangle numbers.

Telescoping Series

Age 16 to 18 Challenge Level:

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

The Harmonic Triangle and Pascal's Triangle

Age 16 to 18

The harmonic triangle is built from fractions with unit numerators using a rule very similar to Pascal's triangle.

Growing

Age 16 to 18 Challenge Level:

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)

Tens

Age 16 to 18 Challenge Level:

When is $7^n + 3^n$ a multiple of 10? Can you prove the result by two different methods?

Summit

Age 16 to 18 Challenge Level:

Prove that the sum from t=0 to m of (-1)^t/t!(m-t)! is zero.

Discrete Trends

Age 16 to 18 Challenge Level:

Find the maximum value of n to the power 1/n and prove that it is a maximum.

Elevens

Age 16 to 18 Challenge Level:

Add powers of 3 and powers of 7 and get multiples of 11.

Binomial

Age 16 to 18 Challenge Level:

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

Bina-ring

Age 16 to 18 Challenge Level:

Investigate powers of numbers of the form (1 + sqrt 2).

Binomial Coefficients

Age 14 to 18

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.

Remainder Hunt

Age 16 to 18 Challenge Level:

What are the possible remainders when the 100-th power of an integer is divided by 125?