What comes next in the sequence \( 1, -3, 9, -27, 81, \dots?\) This sequence is alternating between positive and negative terms, and the first terms are multiples of \(3\). This suggests that we might try taking the ratio of successive terms, which gives

The Lagrange interpolation formula is a way to find a polynomial which takes on certain values at arbitrary points. Specifically, it gives a constructive proof of the theorem below. This theorem can be viewed as a generalization of the well-known fact that two points uniquely determine a straight line, three points uniquely determine the graph of a quadratic polynomial, four points uniquely ...

We have \[\begin{align} \sum _{ i=1 }^{ n }{ 2i } &=2+4+6+\cdots+2n\\ &=2(1+2+3+\cdots+n)\\ &=2\left( \frac { n(n+1) }{ 2 } \right) \\ &=n(n+1).\ _\square \end{align}\]

The limit of a sequence is the value the sequence approaches as the number of terms goes to infinity. Not every sequence has this behavior: those that do are called convergent, while those that don't are called divergent. Limits capture the long-term behavior of a sequence and are thus very useful in bounding them. They also crop up frequently in real analysis. Here, we will …

\[ \frac {1}{3}\pi\times 24^2\times7= 1344\pi.\ _\square\] You must answer a trivia question before the sand in the timer falls to the bottom. The sand falls at a rate of 50 cubic millimeters per second. How much time do you have to answer the question?

A plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analog of a point (zero dimensions), a line (one dimension), and three-dimensional space. A plane in three-dimensional space has the equation ...

A prime number is a natural number greater than 1 that has no positive integer divisors other than 1 and itself. For example, 5 is a prime number because it has no positive divisors other than 1 and 5. In contrast to prime numbers, a composite number is a positive integer greater than 1 that has more than two positive divisors. For example, 4 is a composite number because it has three …

Notice the pattern of the last digits. They are \(7,9,3,1,7,9,3,1,7,9,\ldots\). The last digit repeats in a pattern that is 4 digits long: \(7,9,3,1\).

Suppose \(M\) is an \(n\)-by-\(n\) matrix. When \(M\) has entries in \(\mathbb{C}\), one can prove the Cayley-Hamilton theorem as follows:

4 天之前 · A partition of a positive integer \( n \) is an expression of \( n \) as the sum of one or more positive integers (or parts).