Useful Methods And Tips

$\underline{Sum\ of\ first\ N\ natural\ numbers:}$

The sum of first $N$ natural numbers starting with $1$ can be calculated using the expression:

$Sum = \dfrac{N\times(N+1)}{2}$

For example sum of first ten natural numbers

$1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = \dfrac{10 \times 11}{2} = \dfrac{5 \times 11}{1} = 55$

Likewise:

The sum of first $20$ natural numbers is:

$1 + 2 + 3 ... + 20 = \dfrac{20 \times 21}{2} = \dfrac{10 \times 21}{2} = 210$

$\underline{Identifying\ Simple\ Series:}$

Series of numbers can be formed by many different ways. The simplest ways to form series would be to either add, subtract, multiply the previous number by a fixed number.

Examples:

$4,\ 7,\ 10,\ 13,\ ...$

Here $3$ is added to each number to obtain the next number

$3,\ 6,\ 12,\ 24,\ ...$

Here each number is multiplied by $2$ to obtain the next number.

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$\underline{Identifying\ Complex\ Series}$

A series can be formed by more than one operation performed on the previous number.

Example:

$7,\ 12,\ 22,\ 42,\ ...$

This series can be obtained by multiplying the previous number by $2$ and subtracting $2$ from the result.

$7 \times 2 - 2 = 12$

$12 \times 2 - 2 = 22$

$22 \times 2 - 2 = 42$

$\underline{Combination\ of\ two\ or\ more\ series}$

We can form a series by combining terms from two different series.

$3,\ 4,\ 7,\ 8,\ 11,\ 16,\ 15,\ 32,\ ...$

If you take every odd term of the above series you get:

$3,\ 7,\ 11,\ 15,\ ...$

and the even terms will give you

$4,\ 8,\ 16,\ 32\ ...$

The first series is formed by adding $4$ to the previous term, while the second series is formed by multiplying the previous term by $2$.