`\color{fuchsia} \(★ mathbf\ul"Arithmetic Progression (A.P.)")`

A sequence `color(blue)(a_1, a_2, a_3,…..., a_n,…...)` is called arithmetic sequence or arithmetic progression

if `color(red)(a_(n + 1) = a_n + d) , n ∈ N,` where `color(blue)(a_1)` is called `color(blue)("the first term")` and the constant term `color(blue)(d)` is called `color(blue)("the common difference")` of the `A.P.`

Let us consider an A.P. (in its standard form) with first term a and common difference `d`, i.e., `a, a + d, a + 2d, ......`

Then the `n^(th)` term (`color(blue)("general term")`) of the `A.P. ` is `color(red)(a_n = a + (n – 1) d.)`

We can verify the following `color(green)("simple properties of an A.P. :")`
`color(green)((i))` If a constant is added to each term of an A.P., the resulting sequence is also an A.P.

`color(green)((ii))` If a constant is subtracted from each term of an A.P., the resulting sequence is also an A.P.

`color(green)((iii))` If each term of an A.P. is multiplied by a constant, then the resulting sequence is also an A.P.

`color(green)((iv))` If each term of an A.P. is divided by a non-zero constant then the resulting sequence is also an A.P.

Here, we shall use the `color(red)("following notations for an arithmetic progression:")`

`color(blue)(a) =` the first term, ` \ \ \ \ \ \color(blue)( l )=` the last term, ` \ \ \ \ \ color(blue)(d) =` common difference,

`color(blue)(n) =` the number of terms.

`color(blue)(S_n)=` the sum to `n` terms of `A.P. `

Let `a, a + d, a + 2d, …, a + (n – 1) d` be an A.P. Then `color(red)(l = a + (n – 1) d)`

`color(red)(S_n=(n/2)[2a+(n-1)d])`

We can also write, `color(red)(S_n=(n/2)[a+l])`

`\color{fuchsia} \(★ mathbf\ul"Arithmetic mean")`

`\color{red} ✍️` Given two numbers `a` and `b. `We can insert a number `A` between them so that `a, A, b` is an `A.P.`

`\color{red} ✍️` Such a number `color(blue)(A)` is called the `color(blue)"arithmetic mean"` (A.M.) of the numbers `a` and `b.`

`\color{red} ✍️` Note that, in this case, we have `A – a = b – A,` i.e., `A = (a + b)/2`

`\color{red} ✍️` We may also interpret the A.M. between two numbers `a` and `b` as their average `( a + b)/2`

`\color{blue} =>` More generally, given any two numbers `a` and `b,` we can insert as many numbers as we like between them such that the resulting sequence is an A.P.

Let `color(blue)(A_1, A_2, A_3, …, A_n)` be `n` numbers between `a` and `b` such that `color(blue)(a, A_1, A_2, A_3, …, A_n, b)` is an A.P.

Here, `b` is the `(n + 2) th` term, i.e., `b = a + [(n + 2) – 1]d = a + (n + 1) d.`

This gives `d=(b-a)/(n+1)`

Thus, `n` numbers between `a` and `b` are as follows:

`A_1 = a + d = a +(b-a)/(n+1)`

`A_2 = a + 2d = a +2 * (b-a)/(n+1)`

`A_3 = a + 3d = a +3 * (b-a)/(n+1)`

`..... ..... ..... .....`
`..... ..... ..... .....`

`A_n = a + nd = a +n * (b-a)/(n+1)`

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