• Create a sequence of rectangles using this rule starting with a (1 times 2 ) rectangle. Then write out the sequence of perimeters for the rectangles (the first term of the sequence would be 6, since the perimeter of a (1 times 2 ) rectangle is 6 - the next term would be 10).

• Repeat the above part this time starting with a (1 times 3 ) rectangle. • Find recursive formulas for each of the sequences of perimeters you found in parts (a) and (b). Don't forget to give the initial conditions as well. • Are the sequences arithmetic? If not, are they close to being either of these (i.e., are the differences or ratios almost constant)? 10 Consider the sequence (2, 7, 15, 26, 40, 57, ldots ) (with (a_0 = 2 )). By looking at the differences between terms, express the sequence as a sequence of partial sums.

Then find a closed formula for the sequence by computing the (n )th partial sum.

A collection of and the corresponding Discrete geometry and combinatorial geometry are branches of that study properties and constructive methods of geometric objects. Most questions in discrete geometry involve or of basic geometric objects, such as,,,,,, and so forth.

Buy Geometry and Discrete Mathematics 12 on Amazon.com ✓ FREE SHIPPING on qualified orders. Math.3336: Discrete Mathematics Primes and Greatest Common Divisors Instructor: Dr. Blerina Xhabli. Discrete Mathematics Primes and Greatest Common Divisors 12/26.

The subject focuses on the combinatorial properties of these objects, such as how they one another, or how they may be arranged to cover a larger object. Discrete geometry has a large overlap with and, and is closely related to subjects such as,,,,,,. Main articles: and Packings, coverings, and tilings are all ways of arranging uniform objects (typically circles, spheres, or tiles) in a regular way on a surface. A sphere packing is an arrangement of non-overlapping within a containing space. The spheres considered are usually all of identical size, and the space is usually three-. However, sphere can be generalised to consider unequal spheres, n-dimensional Euclidean space (where the problem becomes in two dimensions, or packing in higher dimensions) or to spaces such as.

A tessellation of a flat surface is the tiling of a using one or more geometric shapes, called tiles, with no overlaps and no gaps. In, tessellations can be generalized to higher dimensions. Specific topics in this area include: • • • • • • • Structural rigidity and flexibility [ ].

Main articles: and A discrete group is a G equipped with the. With this topology, G becomes a. A discrete subgroup of a topological group G is a H whose is the discrete one. For example, the, Z, form a discrete subgroup of the, R (with the standard ), but the, Q, do not. A lattice in a is a with the property that the has finite. Drajver zapominayuschih ustrojstv dlya usb. In the special case of subgroups of R n, this amounts to the usual geometric notion of a, and both the algebraic structure of lattices and the geometry of the totality of all lattices are relatively well understood.