As we have seen, the power of the form, where a, is called Newton's binomial. Also, when n = 0 we have when n = 1 we have when n = 2 we have when n = 3 we have when n = 4 we have Note that the coefficients of the developments were the Pascal triangle. Then we can also write: In general, when the exponent is n, we can write Newton's binomial development formula: Note that the exponents of a decrease from unit to unit, ranging from n to 0, and the exponents of b increase from unit to unit, ranging from 0 to n.

Five-digit number We know that the fifth digit is half of the fourth and a quarter of the third digit: 5th digit = a The fourth digit is double the fifth: 4th digit = 2a The third digit is the quadruple of the fifth: 3rd digit = 4th The third digit is half of the first and double of the fourth.

Beatriz's early clock We know that every hour the minute hand rotates the clock 360º, ie 6º per minute. As for the hour hand, it rotates 30º (which is the angle between two hour marks) per hour. That is, 1st every 2 minutes, or half a degree per minute.

Any natural number greater than 1 can be decomposed into a product of two or more factors. Decomposition of number 24 into one product: 24 = 4 x 6 24 = 2 x 2 x 6 24 = 2 x 2 x 2 x 3 = 2 3 x 3 In the 2 x 2 x 2 x 3 product, all factors are prime. We call factorization of 24 the decomposition of 24 into a prime factor product.

A series is absolutely convergent if the module series is convergent. For example, the alternating series is absolutely convergent because the module series is a p-series with p = 2> 1 and therefore convergent. Theorem If an infinite series is absolutely convergent, then the series is convergent.