Arithmetic and Geometric Progressions

Arithmetic Progressions (AP) §

Definition: §

An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference and is usually denoted by $d$.

Formula: §

For an arithmetic progression, the $n^{th}$ term $a_n$ is given by: $a_n=a_1+d(n-1)$ Where:

• $a_1$ is the first term.
• $d$ is the common difference.
• $n$ is the term number.

Sum of AP: §

The sum of the first $n$ terms of an arithmetic progression is given by: $S_n=\frac{n}{2}(a_1+a_n)$ or equivalently, $S_n=\frac{n}{2}[2a_1+(n-1)d]$

Observation: §

A sequence is an arithmetic progression precisely when each term is the average, or arithmetic mean, of the preceding and the following term.

Arithmetic Mean §

The arithmetic mean, often simply called the “average,” is the sum of a set of numbers divided by the count of those numbers. For a set of numbers $a_1,a_2,...,a_n$, the arithmetic mean $AM$ is given by: $AM=\frac{a_1+a_2+...+a_n}{n}$

For example, the arithmetic mean of the numbers 2, 4, and 6 is: $AM=\frac{2+4+6}{3}=4$

The arithmetic mean gives a central value of the data set and is influenced by each value in the set.

For a set of numbers $a_1,a_2,...,a_n$, the arithmetic mean $AM$ can also be expressed using the $\sum$ notation as: $AM=\frac{1}{n}{\sum }_{i=1}^n{a}_i$ Where:

• $n$ is the total number of elements in the set.
• $a_i$ represents the ith element of the set.

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Geometric Progressions (GP) §

Definition: §

A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This ratio is usually denoted by $r$.

Formula: §

For a geometric progression, the $n^{th}$ term $a_n$ is given by: $a_n=a_1\times r^{(n-1)}$ Where:

• $a_1$ is the first term.
• $r$ is the common ratio.
• $n$ is the term number.

Sum of GP: §

The sum of the first $n$ terms of a geometric progression is given by: $S_n=\frac{a_1(1-r^n)}{1-r}$ for $r\ne 1$.

Observation: §

A sequence is a geometric progression precisely when each term is the geometric mean of the preceding and the following term.

Geometric Mean §

The geometric mean is the nth root of the product of n numbers. For a set of positive numbers $a_1,a_2,...,a_n$, the geometric mean $GM$ is given by: $GM=\sqrt[n]{a_1\times a_2\times ...\times a_n}$

For example, the geometric mean of the numbers 2, 4, and 8 is: $GM=\sqrt[3]{2\times 4\times 8}=4$

It’s important to note that the geometric mean is only defined for sets of positive numbers. If any number in the set is zero or negative, the geometric mean is not defined (or would be zero in the case of a zero value).

The geometric mean is particularly useful when comparing different products or values that have different units or scales. It tends to give a kind of “average growth rate” when looking at proportional growth over time.

For a set of positive numbers $a_1,a_2,...,a_n$, the geometric mean $GM$ can also be expressed using the $\prod$ notation as: $GM=\sqrt[n]{{\prod }_{i=1}^n{a}_i}$ Where:

• $n$ is the total number of elements in the set.
• $a_i$ represents the ith element of the set.

In the $\prod$ notation, the expression inside represents the product of all the elements from $a_1$ to $a_n$. The nth root is then taken of this product to find the geometric mean.