Number+Sequences

=1. Patterns and Sequences=

Find the next two terms in each of the following.


 * a.** 1, 3, 5, 7, 9, 11,, , ...


 * b.** 5, 10, 20, 40, 80, 160,, , ...


 * c.** A, D, G, J, M, P,, , ...


 * d.** J, F, M, A, M, J,, , ...


 * e.** 2, 3, 5, 7, 11, 13,, , ...


 * f.** 1, 3, 6, 10, 15, 21,, , ...


 * g.** 1, 2, 4, 8, 16, 32,, , ...


 * h.** 1, 3, 9, 27, 81, 243,, , ...


 * i.** 1, 3, 7, 15, 31, 63,, , ...


 * j.** 1, 4, 27, 256, 3125, 46656,, , ...

A **sequence** is an ordered arrangement of numbers, figures, or objects. A sequence has items or terms identified as 1st, 2nd, 3rd, and so on. Often, sequences can be identified by their properties.

= = =2. Arithmetic Sequences / Arithmetic Progression (AP)=

An **arithmetic sequence** is one in which each successive term is obtained from the previous term by the addition or subtraction of a fixed number, the **difference**. For example, the set of even numbers is an arithmetic sequence with a difference of 2.


 * Number of term ||  Term  ||
 * 1 ||  2  ||
 * 2 ||  4  ||
 * 3 ||  6  ||
 * 4 ||  8  ||
 * 5 ||  10  ||
 * n ||  2n  ||
 * n ||  2n  ||
 * n ||  2n  ||
 * n ||  2n  ||

In general, the formula for finding the nth term in an AP is a + (n-1)d.


 * Number of term ||  Term  ||
 * 1 ||  a  ||
 * 2 ||  a + d  ||
 * 3 ||  a + 2d  ||
 * 4 ||  a + 3d  ||
 * 5 ||  a + 4d  ||
 * n ||  a + (n - 1)d  ||
 * n ||  a + (n - 1)d  ||
 * n ||  a + (n - 1)d  ||
 * n ||  a + (n - 1)d  ||

=3. Geometric Sequence / Geometric Progression (GP)=

A child has 2 biological parents, 4 grandparents, 8 great grandparents, 16 great-great grandparents, and so on. The number of ancestors form the **geometric sequence** 2, 4, 6, 16, 32,. . . . Each successive term of a geometric sequence is obtained from its predecessor by multiplying by a fixed number, the **ratio**. In this example, the ratio is 2.


 * Number of term ||  Term  ||
 * 1 ||  2  ||
 * 2 ||  4 = 2^2  ||
 * 3 ||  8 = 2^3  ||
 * 4 ||  16 = 2^4  ||
 * 5 ||  32 = 2^5  ||
 * n ||  2^n  ||
 * n ||  2^n  ||
 * n ||  2^n  ||
 * n ||  2^n  ||

In general, the formula for finding the nth term in an GP is ar^(n-1 ).
 * Number of term ||  Term  ||
 * 1 ||  a  ||
 * 2 ||  a  r  ||
 * 3 ||  a  r^2  ||
 * 4 ||  a  r^3  ||
 * 5 ||  a  r^4  ||
 * n ||  a    r^(n-1 )  ||
 * n ||  a    r^(n-1 )  ||
 * n ||  a    r^(n-1 )  ||
 * n ||  a    r^(n-1 )  ||