Finding the General Term and Summing Arithmetic and Geometric Progressions

Finding the General Term of Arithmetic and Geometric Progressions

A sequence is a list of numbers in a specific order.
An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant.
A geometric progression (GP) on the other hand, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio.

The nth term or the general term of an arithmetic progression where ‘a’ is the first term and ‘d’ is the common difference is given by a + (n - 1)d.
The expression a + (n - 1)d gives the nth term in terms of the first term a, the common difference d, and the position n.

The nth term or the general term of a geometric progression where ‘a’ is the first term and ‘r’ is the common ratio is given by a * r^(n - 1).
The expression a * r^(n - 1) gives the nth term in terms of the first term a, the common ratio r, and the position n.

The sum ‘S’ of an arithmetic progression of ‘n’ terms or arithemetic series, where ‘a’ is the first term and ‘l’ is the last term is given by S = n/2 * (a + l) or S = n/2 * (2a + (n - 1)d)

The sum of a finite geometric progression or geometric series is given by S = a * (1 - r^n) / (1 - r) for r ≠ 1.
The sum of the first ‘n’ terms, S_n, is given by S_n = a * (1 - r^n) / (1 - r) for r ≠ 1, where ‘n’ is the number of terms to be added, ‘a’ is the first term and ‘r’ is the common ratio.
For an infinite geometric progression where the absolute value of the common ratio ‘r’ is less than 1, the sum S of terms can be calculated using the formula S = a / (1 - r).