Sequence-and-series Formula Sheet

A sequence is an ordered list of numbers following a particular pattern, while a series is the sum of the terms in a sequence.

Neetesh Kumar | May 17, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Arithmetic Progression (AP)
• 2. Geometric Progression (GP)
• 3. Harmonic Progression (HP)
• 4. Arithmetic Mean (AM)
• 5. Geometric Mean (GM)
• 6. Harmonic Mean (HM)
• 7. Arithmetic-Geometric Progression (AGP)
• 8. Properties of Sigma Notations
• 9. Some Important Summation series results

1. Arithmetic Progression (AP):

Arithmetic Progression is a sequence whose terms increase or decrease by a fixed number. This fixed number is called the common difference. If ‘a’ is the first term & ‘d’ is a common difference, then Arithmetic Progression can be written as a, a + d, a + 2d, ..., a + (n – 1) d, ...

• We can find n$^{th}$ term of an Arithmetic Progression as $\fcolorbox{red}{yellow}{$T_n = a + (n-1)d$}$ where d = $T_n - T_{n-1}$
  
  
• The Sum of the first n terms : $\fcolorbox{red}{yellow}{$S_n = \frac{n}{2}[2a + (n-1)d] \space or \space S_n= \frac{n}{2}[a + l]$}$ where $l$ is the last term.
  
  
• We can also write n$^{th}$ term as $\fcolorbox{red}{yellow}{$T_n = S_n - S_{n-1}$}$

$\bold{Things \space to \space Note:}$

• n$^{th}$ term of an Arithmetic Progression is of the form An + B, i.e. a linear expression in n, in such case, the coefficient of n is the common difference of the A.P., i.e. A
• Sum of the first n terms of an Arithmetic Progression is An$^2$ + Bn, i.e., a quadratic expression in n. In such case, the common difference is twice the coefficient of n$^2$, i.e., 2A
• Three numbers in Arithmetic Progression can be taken as $\bold{a – d, a, a + d}$;
  Four numbers in Arithmetic Progression can be taken as $\bold{a – 3d, a – d, a + d, a + 3d}$
  Five numbers in Arithmetic Progression are $\bold{a – 2d, a – d, a, a + d, a + 2d}$
  Six terms in Arithmetic Progression are $\bold{a – 5d, a – 3d, a – d, a + d, a + 3d, a + 5d}$ etc.
• If a, b, c are in Arithmetic Progression, then b = $\frac{a+c}{2}$
• If $a_1, a_2, a_3, ....$ and $b_1, b_2, b_3,.....$ are two A.P.s, the sum and difference of these Arithmetic Progression is $(a_1 \plusmn b_1, a_2 \plusmn b_2, a_3 \plusmn b_3, .....)$ are also in Arithmetic Progression.
• Any term of an Arithmetic Progression (except the first & last) equals half the sum of terms that are equidistant from it. $T_r = \frac{T_{r-k} + T_{r+k}}{2},$ k < r
• If each term of an Arithmetic Progression is increased or decreased by the same number, then the resulting sequence is also an Arithmetic Progression having the same common difference.
• If each term of an Arithmetic Progression is multiplied or divided by the same nonzero number (k), then the resulting sequence is also an Arithmetic Progression whose common difference is kd & $\frac{d}{k}$ respectively, where d is the common difference of original Arithmetic Progression.

2. Geometric Progression (GP):

Geometric Progression is a sequence of numbers whose first term is non-zero & each of the succeeding terms equals the preceding terms multiplied by a constant. Thus, the ratio of successive terms is constant in a Geometric Progression.

This constant factor is called the common ratio of the series & is obtained by dividing any term by the immediately previous term. Therefore $a, ar, ar^2, ar^3, ar^4, .....$ is a Geometric Progression with 'a' as the first term & 'r' as a common ratio.

• We can find n$^{th}$ term of an Geometric Progression as $\fcolorbox{red}{yellow}{$T_n = ar^{n-1}$}$
  
  
• Sum of the first n terms $\fcolorbox{red}{yellow}{$S_n = \frac{a(r^n - 1)}{(r-1)}, \space if \space r \ne 1$}$
  
  
• Sum of infinite terms of a Geometric Progression is $\fcolorbox{red}{yellow}{$S_\infty = \frac{a}{1-r}$}$ when |r| < 1 (n $\rightarrow \infty, r^n \rightarrow 0$)
  
  
• If a, b, c are in Geometric Progression $\Rightarrow b^2 = ac \Rightarrow$ loga, logb, logc, are in Arithmetic Progression.

$\bold{Things \space to \space Note:}$

• In a Geometric Progression, the product of the k$^{th}$ term from the beginning and the k$^{th}$ term from the last is always constant, equal to the product of the first and last term.
• Three numbers in Geometric Progression : $\bold{\frac{a}{r}, a , ar}$
  Four numbers in Geometric Progression : $\bold{\frac{a}{r^3}, \frac{a}{r}, ar , ar^3}$
  Five numbers in Geometric Progression : $\bold{\frac{a}{r^2}, \frac{a}{r}, a , ar, ar^2}$
  Six numbers in Geometric Progression : $\bold{\frac{a}{r^5}, \frac{a}{r^3}, \frac{a}{r}, ar , ar^3, ar^5}$
  
• If each term of a Geometric Progression is raised to the same power, then the resulting series is also a Geometric Progression.
• If each term of a Geometric Progression is multiplied or divided by the same non-zero quantity, then the resulting sequence is also a Geometric Progression.
• If $a_1, a_2, a_3, ...$ and $b_1, b_2, b_3, ...$ be two geometric progressions of common ratio $r_1$ and $r_2$ respectively, then $a_1b_1, a_2b_2, ...$ and $\frac{a_1}{b_1}, \frac{a_2}{b_2}, \frac{a_3}{b_3},...$ will also form a Geometric Progression whose common ratio will be $r_1, r_2$ and $\frac{r_1}{r_2}$ respectively.
• In a positive Geometric Progression, every term (except the first) is equal to the square root of the product of its two terms, which are equidistant from it, i.e., $T_r = \sqrt{T_{r-k}T_{r+k}}$, k < r
• If $a_1, a_2, a_3, ..., a_n$ is a Geometric Progression of nonzero, nonnegative terms then $loga_1, loga_2, loga_2, ..., loga_n$ is an Arithmetic progression and vice-versa.

3. Harmonic Progression (HP):

A sequence is said to be in Harmonic Progression if the reciprocals of its terms are in Arithmetic Progression. If the sequence $a_1, a_2, a_3, ..., a_n$ is a Harmonic Progression then $\frac{1}{a_1}, \frac{1}{a_2},..., \frac{1}{a_n}$ Arithmetic Progression & vice-versa.

Here, we have the formula to find the n$^{th}$ term but do not have the formula for the sum of the first n terms of a Harmonic Progression.
The general form of a harmonic progression is $\frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2d}, ....., \frac{1}{a+(n-1)d}$

$\bold{Things \space to \space Note:}$

• No term of any Harmonic Progression can be zero.
• If a, b, c are in Harmonic Progression $\Rightarrow b = \frac{2ac}{a+c}$ or $\frac{a}{c} = \frac{a-b}{b-c}$

4. Arithmetic Mean (AM):

If three terms are in Arithmetic Progression, then the middle term is the Arithmetic Mean between the other two, so if a, b, and c are in Arithmetic Progression, b is AM of a & c.

n arithmetic means between two numbers:
If a, b are any two given numbers & a, $A_1, A_2, ...., A_n, b$ are in Arithmetic progression then $A_1, A_2, ...., A_n$ are the n AM’s between a & b, then $A_1 = a+d, A_2 = a+2d, ...., A_n = a+nd$ where $d = \frac{b-a}{n+1}$

$\bold{Things \space to \space Note:}$

• Sum of n AM's inserted between a & b is equal to n times the single AM between a & b, i.e.

$$\sum_{r=1}^{n} A_r = nA$$

where A is the single AM between a & b i.e., $\frac{a+b}{2}$

5. Geometric Mean (GM):

If a, b, and c are in Geometric Progression, then b is the Geometric Mean between a & c i.e., b$^2$ = ac, therefore b = $\sqrt{ac}$

n Geometric Mean (GM) means between two numbers:
If a, b are any two given positive numbers & a, $G_1, G_2, ...., G_n, b$ are in Geometric progression then $G_1, G_2, ...., G_n$ are the n GM’s between a & b, then $G_1 = ar, G_2 = ar^2, ...., G_n = ar^n$ where $r = (\frac{b}{a})^\frac{1}{n+1}$

$\bold{Things \space to \space Note:}$

• The product of n GMs between a & b is equal to n$^{th}$ power of the single GM between a & b i.e. $\Pi_{r=1}^{n}G_r = (G)^n$ where G is the single GM between a & b i.e. $\sqrt{ab}$

6. Harmonic Mean (HM):

If a, b, and c are in HP, then b is HM between a & c, then $b = \frac{2ac}{a+c}$

$\bold{Things \space to \space Note:}$
If A, G, and H are, respectively, AM, GM, and HM between two positive numbers a & b, then
$\bold{(i)}$

• $G^2 = AH$ (A, G, H constitute a GP)
• $A \ge G \ge H$
• $A = G = H \iff a = b$
  

$\bold{(ii)}$
Let $a_1, a_2, a_3, ..., a_n$ be n positive real numbers, then we define their
Arithmetic mean as A = $\frac{a_1+a_2+...+a_n}{n}$
Geometric mean as G = $(a_1.a_2...a_n)^\frac{1}{n}$
Harmonic mean as H = $\frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + ...+ \frac{1}{a_n}}$
It can be shown that $A \ge G \ge H$ Moreover, equality holds at either place $\iff a_1 = a_2 = ... = a_n$

7. Arithmetic-Geometric Progression (AGP):

Sum of First n terms of an Arithmetic-Geometric Progression
Let $\space$ $S_n = a + (a+d)r + (a+2d)r^2 + .... + [a+(n-1)d]r^{n-1}$
then $S_n = \frac{a}{1-r} + \frac{dr(1-r^{n-1})}{(1-r)^2} - \frac{[a+(n-1)d]r^{n-1}}{1-r}, r \ne 1$
Sum to Infinity
If |r| < 1 and $n \rightarrow \infty$ then $Lim_{n \rightarrow \infty} r^n = 0 \Rightarrow S_\infty = \frac{a}{1-r} + \frac{dr}{(1-r)^2}$

8. Properties of Sigma Notations:

$\bold{(a)}$ $\sum_{r=1}^{n} (a_r \plusmn b_r) = \sum_{r=1}^{n} a_r \plusmn \sum_{r=1}^{n} b_r$
$\bold{(a)}$ $\sum_{r=1}^{n} ka_r = k\sum_{r=1}^{n} a_r$
$\bold{(a)}$ $\sum_{r=1}^{n} k = nk;$ where k is a constant.

9. Some Important Summation series results:

$\bold{(a)}$ Sum of the first n natural numbers is = $\sum_{r=1}^{n} r = \frac{n(n+1)}{2}$
$\bold{(b)}$ Sum of the squares of the first n natural numbers is = $\sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6}$
$\bold{(c)}$ Sum of the cubes of the first n natural numbers is = $\sum_{r=1}^{n} r^3 = \frac{n^2(n+1)^2}{4}$
$\bold{(a)}$ $\sum_{r=1}^{n} r^4 = \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$

$\color{red} \bold{Related \space Pages:}$
Sequence and Series Calculators
Operation on Matrices
Matrices Formula Sheet