Arithmetic Progression (AP)

General term: $$a_n=a_1+(n-1)d$$

Sum of n terms: $$S_n=\frac{n}{2}(a_1+a_n)$$

Properties:

1. The difference between two consecutive terms of an AP is constant.
2. The sum of the first n odd natural numbers is a perfect square.
3. The sum of the first n even natural numbers is n times the (n+1)th odd natural number.

Geometric Progression (GP)

General Term:-$$T_n=ar^{n-1}$$ Sum of n terms:-$$S_n=\frac{a(r^n-1)}{r-1}, where r\neq1$$

Properties:

1. The ratio between two consecutive terms of a GP is constant.
2. The sum of the first n terms of a GP with common ratio r and first term a is ( \frac{a(1-r^n)}{1-r} ), where (r \ne 1).
3. If (0 < r < 1), then the sum of the infinite geometric series (a+ar+ar^2+\dotsb) is ( \frac{a}{1-r},).

Harmonic Progression (HP)

General term: $$h_n=\frac{1}{a+nd}$$

Sum of n terms: $$S_n= \frac{n}{2a+(n-1)d}$$

Series

Convergent series: A series is said to be convergent if it has a finite number of the sums.

Divergent series: A series that doesn’t converge is called a divergent series.

Alternating series: An alternating series is a series whose terms alternate in sign.

Mathematical induction: Mathematical induction is a method of proving that a statement holds for all natural numbers.

Limits of sequences

Definition of limit: The limit of a sequence ((a_n)) is the number L if, for any (\varepsilon>0), there exists an integer (N) such that (|a_n - L|< \varepsilon) whenever (n>N).

Properties of limits:

1. If (\lim\limits_{n\to \infty}a_n = L) and (\lim\limits_{n\to\infty}b_n = M,) then a) (\lim\limits_{n\to\infty} (a_n +b_n) = L + M). b) (\lim\limits_{n\to\infty} (a_n - b_n) = L - M). c) (\lim\limits_{n\to\infty} (ca_n) = cL, where c is a constant). d) If (c\neq0), (\lim\limits_{n\to\infty} \frac{a_n}{b_n}=\frac{L}{M}).

2. If (\lim\limits_{n\to \infty}a_n = L), then for every (\varepsilon > 0), there exists an integer (N) such that (|a_n-L|\leq\varepsilon) for all (n>N).

Limit theorems:

1. Squeeze theorem: If (a_n \le b_n \le c_n) for all (n) and if (\lim\limits_{n\to \infty} a_n = \lim\limits_{n\to \infty} c_n = L,) then (\lim\limits_{n\to \infty} b_n = L.)

2. L’Hopital’s rule: If (\lim\limits_{n\to a}f(x) = \lim\limits_{n\to a} g(x) = 0) or (\pm \infty), and if (\lim\limits_{n\to a} \frac{f’(x)}{g’(x)}) exists, then (\lim\limits_{n\to a} \frac{f(x)}{g(x)} = \lim\limits_{n\to a} \frac{f’(x)}{g’(x)}).

Applications of sequences and series

• Approximating functions: Sequences and series can be used to approximate functions. For example, the Maclaurin series for (e^x) is $$e^x = \sum\limits_{n=0}^{\infty}\frac{x^n}{n!} = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+ \dots$$ This series can be used to approximate the value of (e^x) for any real number (x).

• Calculating sums: Sequences and series can be used to calculate the sums of infinite series. For example, the sum of the series $$\sum\limits_{n=1}^{\infty} \frac{1}{n^2} = \frac{1}{1}+\frac{1}{2^2}+\frac{1}{3^2}+\dots$$ can be calculated using the formula $$\sum\limits_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$

• Solving equations: Sequences and series can be used to solve equations. For example, the equation (x = e^x) can be solved using the iterative method: $$x_{n+1} = e^{x_n}$$ Starting with an initial guess (x_0), we can use this formula to generate a sequence of approximations that converge to the solution of the equation.