## Progressions and Series

Theory

Progressions and Series is one of the three important topics in the quantitative section in CAT and a significant number of questions appear in the examination from this section every year. Some of the questions from this section can be very tough and time consuming while the others can be very easy. The trick to ace this section is to quickly figure out whether a question is solvable or not and not waste time on very difficult questions. Some of the questions in this section can be answered by ruling out wrong choices among the options available. This method will both save time and improve accuracy. There are many shortcuts which will be of vital importance in answering this section. We will try and upload a few to help students.

Formula
• 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 = $$\frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+...+\frac{1}{x_{n}}}$$
Theory
• Arithmetic Mean $$\geq$$ Geometric mean $$\geq$$ Harmonic Mean
• The equality holds true if and only if all the terms are equal.
Theory

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

Theory

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)

Theory

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
Formula

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 $$T_{n}=S_{n}-S_{n-1}$$
Formula

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= $$\frac{2ab}{a+b}$$
Formula

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)

Formula

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, $$ar^{2}$$,...
Formula

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,....
Formula
• 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 $$\frac{a}{1-r}$$
Formula

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