• Arithmetic Mean = $$ \frac{x_{1}+x_{2}+x_{3}…x_{n}}{n}$$
• Geometric Mean = $$\sqrt[n]{x_{1}*x_{2}*x_{3}…x_{n}}$$
• Harmonic Mean = $$\frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+...+\frac{1}{x_{n}}}$$

Arithmetic progression (A.P)

If the sum of the difference between any two consecutive terms is constant then the terms are said to be in A.P

Example: 2,5,8,11 or a, a+d, a+2d, a+3d...

If 'a' is the first term and 'd' is a common difference then the general 'n' term is $$T_{n}$$=a+(n-1)d

Sum of first 'n' terms in A.P=$$\frac{n}{2}$$[2a+(n-1)d]

Number of terms in A.P=$$\frac{Last  Term-First  Term}{Common  Difference}$$+1

Properties of Arithmetic progression

If a, b, c, d,.... are in A.P and ‘k’ is a constant then

• a-k, b-k, c-k,... will also be in A.P

• ak, bk, ck,...will also be in A.P

• a/k, b/k, c/k will also be in A.P

Harmonic Progression

• If a, b, c, d,......are unequal numbers then they are said to be in H.P if 1/a, 1/b, 1/c,......are in A.P

• The ‘n’ term in H.P is 1/(nth term in A.P)

Properties of H.P :

If a, b, c, d,...are in H.P, then

                                               a+d > b+c

                                                 ad > bc

Geometric Progression

• If in a succession of numbers the ratio of any term and the previous term is constant then that numbers are said to be in Geometric Progression.
• Ex :1, 3, 9, 27 or a, ar, a$$r^{2}$$, a$$r^{3}$$
• The general expression of a G.P, Tn = a $$r^{n-1}$$ (where a is the first term and ‘r’ is the common ratio).
• Sum of ‘n’ terms in G.P, Sn = $$\frac{a(1-r^{n})}{1-r}$$ (if r<1) or $$\frac {a(r^{n}-1)}{r-1}$$ (if r>1)

Properties of G.P

If a, b , c, d,.... are in G.P and ‘k’ is a constant then

1. ak, bk, ck,...will also be in G.P
2. a/k, b/k, c/k will also be in G.P

Sum of term of infinite series in G.P, $$S_{∞}$$=$$\frac {a}{1-r}$$ (-1 < r <1)

• Arithmetic Mean $$\geq$$ Geometric mean $$\geq$$ Harmonic Mean
• The equality holds true if and only if all the terms are equal.

Sum of the first n terms:

• Sum of the first n natural numbers is $$\frac{n(n+1)}{2}$$
• Sum of the squares of the first n natural numbers is $$\frac{n(n+1)(2n+1)}{6}$$
• The sum of cubes of first 'n' natural numbers = $$(\frac{n(n+1)}{2})^{2}$$
• The sum of first 'n' odd natural numbers= $$n^{2}$$
• The sum of first 'n' even natural numbers= n(n+1)
• In any series, if the sum of first n terms is given by $$S_{n}$$,then the $$n^{th}$$ term $$T_{n}=S_{n}-S_{n-1}$$

Geometric Mean

• If a, b, c,...n terms are in G.P then G.M=$$\sqrt[n]{a \times b \times c \times...n  terms}$$
• If two numbers a,b are in G.P then their G.M= $$\sqrt{a \times b}$$
• Inserting 'n' means between two quantities a and b with common ration 'r'
• Then the number of terms are n+2 and a, b are the first and last terms
• $$r^{n+1}=\frac{b}{a}$$ or r=$$\frac{\sqrt[n+1]{b}}{a}$$
• The final series is a, ar, $$ar^{2}$$,...

Arithmetic mean

• The arithmetic mean = $$\frac{Sum  of  all  the  terms}{Number  of  terms}$$
• If two numbers A and B are in A.P then arithmetic mean= $$\frac{a+b}{2}$$
• Inserting 'n' means between two numbers a and b.
• The total terms will become n+2, a is the first term and b is the last term
• Then the common difference d= $$\frac{b-a}{n+1}$$
• The last term b=a+(n+1)d
• The final series is a, a+d, a+2d,....
  

Arithmetic Geometric Series

• A series will be an arithmetic-geometric series if each of its terms is formed by the product of the corresponding terms of an A.P and G.P.

• The general form of A.G.P series is a, (a+d)r, (a+2d)$$r^{2}$$,......

• Sum of ‘n’ terms of A.G.P series

                                           $$S_{n}$$=$$\frac{a}{1-r}$$+rd$$\frac{(1-r^{n-1})}{1-r}$$+rn$$\frac{[a+(n-1)d]}{1-r}$$(r≠1)

• Sum of infinite terms of A.G.P series

                                          $$S_{∞}$$=$$\frac{a}{1-r}+\frac{dr}{(1-r)^{2}}$$(|r|<1)

Harmonic Mean

• If a, b, c, d...are the given numbers in H.P then the Harmonic mean of 'n' terms=$$\frac{Number  of  terms}{\frac{1}{a}+\frac{1}{a}+\frac{1}{a}+....}$$
• If two numbers a and b are in H.P then the Harmonic mean= $$\frac{2ab}{a+b}$$

• Sum of the first 'n' terms of a Geometric Progression is $$\frac{a(r^{n}-1)}{r-1}$$
• Sum of an infinite geometric progression is $$\frac{a}{1-r}$$

Relationship between AM, GM and HM for two numbers a and b,

• A.M=$$\frac{a+b}{2}$$
• G.M=$$\sqrt{a \times b}$$
• H.M=$$\frac{2ab}{a+b}$$
• G.M=$$\sqrt{AM \times HM}$$
• A.M ≥ G.M ≥ H.M