# Sequence and Series Formulas

Sequence and Series Formulas

A sequence is a ordered list of numbers and series is the sum of the term of sequence.

Some of the important formulas of sequence and series are given below:-

Arithmetic Series

1. For the numbers in arithmetic progression,

N’th terms: $u_n = a +(n-1) d$

Where a = first terms, d = common difference.

N’th mean: $X_n = a + \dfrac{n(b-a)}{n-1}$

2.  For the arithmetic series,

The last term $(1) = a + (n-1)d$

The sum, $S_n = \dfrac{1}{2}n \{2a + (n-1)d \} = \dfrac{1}{2} n (a+1)$

3. Sum of squares of n natural numbers $S_n = \dfrac{n(n + 1)(2n + 1)}{6}$

4. Sum of cubes of n natural numbers $S_n = \{ \dfrac{n(n + 1)}{2} \}^2$

5. For the numbers in geometric progression,

N’th term: $u_n = ar^{n-1}$ and n’th mean, $X_n = a \left( \dfrac{b}{a} \right)^{\dfrac{n}{n + 1}}$

Geometric Series

1. For the geometric series, the sum

$S_n = \dfrac{a ( r^n - 1)}{r-1}$ if r > 1, $S_n =\dfrac{a (1 - r^n)}{1 - r}$ if r < 1

$S_n = \dfrac{a - rl}{1 - r}$ if, r < 1, $S_n = \dfrac{a}{1 - r}$ if  n is very large,

2. For any two numbers a and b.

$\dfrac{a + b}{2} \ge \sqrt{ab}$ if $a \neq b$, $\dfrac{a + b}{2} = \sqrt{ab}$ if a = b

3. Mean

i. a.  Arithmetic mean (A.M.) = $\dfrac{a + b}{2}$

b. For more arithmetic means “d” is to be found as $d = \dfrac{b - a}{n + 1}$  then,

mean

$M_1 = a + d \, \, M_2 = a + 2d \, \,M_3 = a + 3d$ $\cdots \, \, \cdots \, \,\cdots \, \, \cdots$

Where a = first term, b = last term, d = common difference

ii.a. Geometric mean (G.M) = $\sqrt{ab}$

b. For more geometri means “r” is to be found as

$r = \left( \dfrac{b}{a} \right)^{\dfrac{1}{n + 1}}$ then, mean

$M_1 = ar \, \, M_2 = ar^2 \, \, M_3 = ar^3 \, \, \cdots \, \, \cdots$

Where a = first term, b = last term, r = common ratio.

4. If AM & GM be the A.M. A\and G.M. between two positive numbers and b then $AM \ge GM$.

5. Sum of the first ‘n’ natural numbers ( 1+ +2 +3 + ……………………+ n) is $S_n = \dfrac{n}{2} (n+1)$.

6. Sum of the first ‘n’ odd natural numbers (1 + 3 + 5 + ………………………to n terms) is $S_n = n^2$

7. Sum of the first ‘n’ even natural numbers (2 +4 +6 + ……………………….to n terms ) is $S_n = n(n + 1)$

8. Sum of squares of the first ‘n’ natural numbers

$(1^2 + 2^2 + 3^2 +\cdots n^2)$ is $S_n = \dfrac{1}{6}n(n + 1) (2n + 1)$.

9. Sum of the cubes of first ‘n’ natural numbers

$(1^3 + 2^3 + 3^3 + \cdots + n^3)$ is $S_n = \{\dfrac{n(n + 1)}{2} \}^2$.

10. $r = \dfrac{t_n}{t_{n - 1}}$

11. $t_n = ar ^{n - 1}$

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