Number Series and Sequences

 

Number Series and Sequences

 

A set of numbers which increases/decreases under some specific prior decided rules in a particular sequence are known as series.

One has to identify that particular pattern in which that number series is arranged. Each number in a number series is called a term.

There mainly two types of problems based on number series

I.          In a given number series, one of the term will be missing. One has to identify the pattern of the series and find that missing term.

II.         In a given number series, one of the term misfits the pattern or sequence of that series. One has to find that misfit term by identifying the pattern of that series. This process is also known as 'odd one out’.

 

BASIC CLASSIFICATION OF NUMBER SERIES

Increasing Series

In this type of number series, each successive term

Increases, following a certain sequence

There are two types of increasing series

I.          Difference Series Here, each successive term is found by adding some number in a particular pattern.

II.         Ratio Series Here, each successive term is found by multiplying some number in a particular pattern.

 

Decreasing Series

In this type of number series, each successive term decreases, following a certain sequence.

There are two types of decreasing series

 

I.          Difference Series Here, each successive term is found by subtracting some numbers in a particular pattern.

II.         Ratio Series Here, each successive term is found by dividing the previous number in a particular pattern.

 

SERIES (PROGRESSION)

Arithmetic Progression

\[a,\]\[(a+d),\]\[(a+2\,d),\]\[(a+3\,d),...\text{ }\]

a = 1st term, d = common difference. Then,

·         nth term \[=a+(n-1)\,d\]

·         Sum of n terms \[=\frac{n}{2}[(2a+(n-1)d]\]

·         Sum of n terms \[=\frac{n}{2}(a+l),\] where \[l=\] last term

 

Geometric Progression

\[a,\]\[ar,\]\[a{{r}^{2}},\]\[a{{r}^{3}},...\]

\[a=\] 1st term,\[r=\] common ratio. Then,

·         nth term \[=a{{r}^{n-1}}\]

·         Sum of n terms \[=\frac{a\,\,(1-{{r}^{n}})}{(1-r)},\] where \[r<1\]

·         Sum of n terms \[=\frac{a\,\,({{r}^{n}}-1)}{r-1)},\] where \[r>1\]

Formulae on Series

·         Sum of first n natural numbers \[=\frac{n\,\,(n+1)}{2}\]

·         Sum of first n odd numbers \[={{n}^{2}}\]

·         Sum of first n even numbers \[=n\,\,(n+1)\]

·         Sum of squares of first n natural numbers

\[=\frac{n\,\,(n+1)(2n+1)}{6}\]

·         Sum of cubes of first n natural numbers

\[={{\left[ \frac{n\,\,(n+1)}{2} \right]}^{2}}\]

Some of the very basic series are given below

 

Prime Number Series

In these types of series, a series is made by using prime numbers and arranging them in different patterns.

e.g., Find out the next term in the series.

7, 11, 13, 17, 19,?

Sol.      Given series is a consecutive prime number series. Therefore, next term in series will be 23.

 

Multiple Series

In these types, series proceeds as a multiple of a specific number.

e.g., Find out the missing term in the series.

4, 8, 16, 32, 64, ?, 256

Sol.      Here, every next number is double the previous number.

\[\therefore \]      Required number \[=64\times 2=128\]

 

Division Series

Similar to multiple series but here in place of multiples, numbers in series are divided by some specific number.

e.g., Find out the missing term in the series.

80, 200, 500, ?, 3125

Sol.      \[80\times \frac{5}{2}=200,\]\[200\times \frac{5}{2}=500\]

\[500\times \frac{5}{2}=1250,\]\[1250\times \frac{5}{2}=3125\]

Hence, missing term is 1250.

 

Difference or Addition Series

In these types, the pattern followed is of adding or subtracting a specific number.

e.g.. Find out the missing term in the series.

108, 99, 90, 81, ?, 63

Sol.      Here, every next number is 9 less than the previous number.

\[\therefore \] Required number \[=81-9=72\]

 

\[{{\mathbf{n}}^{\mathbf{2}}}\]Series

The pattern followed in this type of series is squaring the number in a specific pattern.

e.g., Find out the missing term in the series.

4, 16, 36, 64, ?, 144

Sol.      This is a series of squares of consecutive even numbers.

i.e.,       \[{{2}^{2}}=4\]

\[{{4}^{2}}=16\]

\[{{6}^{2}}=36\]

\[{{8}^{2}}=64\]

\[{{10}^{2}}=[100]\]

\[{{12}^{2}}=144\]

Hence, missing term is 100.

\[\mathbf{(}{{\mathbf{n}}^{\mathbf{2}}}\mathbf{+1)}\] Series

Very similar to \[{{n}^{2}}\] series but here 1 is added to the square term to form the series.

e.g.. Find out the missing term in the series.

10, 17, 26, 37, ?, 65

Sol.      Series pattern is  \[{{3}^{2}}+1=10;\]   \[{{4}^{2}}+1=17\]

\[{{5}^{2}}+1=26;\]   \[{{6}^{2}}+1=37\]

\[{{7}^{2}}+1=[50]\]

\[{{8}^{2}}+1=65\]

\[\therefore \]      Required number \[=50\]

 

\[\mathbf{(}{{\mathbf{n}}^{\mathbf{2}}}-\mathbf{1)}\] Series

In this series 1 is subtracted from the square number like 1 was subtracted in \[({{n}^{2}}-1)\] series.

e.g., Find out the missing term in the series.

0, 3, 8, 15, 24, ?, 48

Sol.      Series pattern is

\[{{1}^{2}}-1=0\]

\[{{2}^{2}}-1=3\]

\[{{3}^{2}}-1=8\]

\[{{4}^{2}}-1=15\]

\[{{5}^{2}}-1=24\]

\[{{7}^{2}}-1=48\]

\[\therefore \]        Required number \[=35\]

 

\[\mathbf{(}{{\mathbf{n}}^{\mathbf{2}}}\mathbf{+n)}\] Series

In this series the same number is added to its square term to form series.

e.g.. Find out the missing term in the series.

420, 930, 1640, ?, 3660

Sol.      Series pattern is

\[{{20}^{2}}+20,\]\[{{30}^{2}}+30,\]\[{{40}^{2}}+40,\]\[{{50}^{2}}+50,\]            \[{{60}^{2}}+60\]

\[\therefore \]      Required number \[=502+50=2550\]

 

\[\mathbf{(}{{\mathbf{n}}^{\mathbf{2}}}-\mathbf{n)}\] Series

Similar to \[({{n}^{2}}+n)\] series but in this type the specific number is subtracted from the square of this specific number.

e.g.. Find out the missing term in the series.

210, 240, 272, 306, ?, 380

Sol.      Series pattern is

\[{{15}^{2}}-15,\]\[{{16}^{2}}-16,\]\[{{17}^{2}}-17,\]\[{{18}^{2}}-18,\]\[{{19}^{2}}-19,\]            \[{{20}^{2}}-20\]

\[\therefore \]      Required number \[={{19}^{2}}-19=342\]

 

\[{{\mathbf{n}}^{\mathbf{3}}}\] Series

In this type of series the cube of numbers are taken to from the series as in n2 series square of numbers were taken.

e.g., Find out the missing term in the series.

1000, 8000, 27000, 64000, 125000, ?

Sol.      Series pattern is \[{{10}^{3}},\]\[{{20}^{3}},\]\[{{30}^{3}},\]\[{{40}^{3}},\]\[{{50}^{3}},\]\[{{60}^{3}}\]

\[\therefore \]      Required number \[={{60}^{3}}=216000\]

 

\[\mathbf{(}{{\mathbf{n}}^{\mathbf{3}}}\mathbf{+1)}\] Series

This series is formed by adding 1 to each cube number of the series or pattern.

 

e.g., Find out the missing term in the series.

126, 217, 344, ?, 730

Sol.      Series pattern is \[{{5}^{3}}+1,\]\[{{6}^{3}}+1,\]\[{{7}^{3}}+1,\]\[{{8}^{3}}+1,\]\[{{9}^{3}}+1\]

\[\therefore \]      Required number \[={{8}^{3}}+1=513\]

 

\[\mathbf{(}{{\mathbf{n}}^{\mathbf{3}}}-\mathbf{1)}\] Series

Like in \[({{n}^{3}}+1)\]  series 1 was added to form the series. Here, 1 is subtracted from the cube of numbers for forming the series.

e.g., Find out the missing term in the series.

?, 7999, 26999, 63999, 124999

Sol.      Series pattern is

\[{{10}^{3}}-1,\]\[{{20}^{3}}-1,\]\[{{30}^{3}}-1,\]\[{{40}^{3}}-1,\]\[{{50}^{3}}-1\]

\[\therefore \]      Required number \[={{10}^{3}}-1=999\]

 

\[\mathbf{(}{{\mathbf{n}}^{\mathbf{2}}}\mathbf{+n)}\] Series

In this type of series, the specific or particular number itself

is added to the cube of that particular number.

e.g., Find out the missing term in the series.

?, 8020, 27030, 64040

Sol.      Series pattern is

            \[{{10}^{3}}+10,\]\[{{20}^{3}}+20,\]\[{{30}^{3}}+30,\]\[{{40}^{3}}+40\]

\[\therefore \]      Required number \[={{10}^{3}}+10=1010\]

 

\[\mathbf{(}{{\mathbf{n}}^{\mathbf{3}}}-\mathbf{n)}\]Series

Very similar to \[({{n}^{3}}-n)\] series but here the specific number is subtracted in place of addition from the cube of that particular number.

e.g., Find out the missing term in the series.

            ?, 7980, 26970, 63960, 124950

Sol.      Series pattern

\[{{10}^{3}}-10,\]\[{{20}^{3}}-20,\]\[{{30}^{3}}-30,\]\[{{40}^{3}}-40,\]\[{{50}^{3}}-50\]

\[\therefore \]      Required number \[={{10}^{3}}-10=990\]