## HOW TO FIND THE SUM OF SPECIAL SERIES

There are some series whose sum can be expressed by explicit formulae. Such series are called special series.

Here we study some common special series like

(i)  Sum of first "n" natural numbers.

=  n(n + 1)/2

(ii)  Sum of first "n" odd natural numbers.

=  n2  (or)  [(l + 1)/2]2

(iii)  Sum of squares of "n" natural numbers :

=  n(n + 1) (2n + 1) / 6

(iv)  Sum of cubes of "n" natural numbers :

=  [n(n + 1) / 2]2

Note :

We can use the formulas given above only if the given series starts from 1.

If it doesn't start from 1, we have to follow the steps given below to find the required sum.

Step 1 :

Find the sum from 1 upto the last term we have in the given series.

Step 2 :

Find the sum from 1 upto the preceding term of the first term what we have in the given series.

Step 3 :

By subtracting the results from (1) and (2), we will get the required sum. That is,

Required sum  =  Result from step 1 - Result from step (2)

Example 1 :

Find the sum of first 75 natural numbers.

Solution :

Sum of first 75 natural numbers

1 + 2 + 3 + 4 + ................. + 75

Sum of natural numbers  =  n(n + 1)/2

Here "n" stands for total number of terms.

=  75 (75 + 1)/2

=  (75  76) / 2

=  75  38

=  2850

Therefore the sum of first 75 natural number is 2850.

Example 2 :

Find the sum of

15 + 16 + 17 + .............. + 80

Solution :

Since the given series doesn't start from 1, we are decomposing the given series into two parts.

Step 1 :

Find the sum of the series from 1 upto the last term 80.

(1 + 2 + 3 +........+ 80)

Sum of natural numbers  =  n(n + 1) / 2

n  =  80

=  80(80 + 1) / 2

=  40 ⋅ 81

=  3240  -----(1)

Step 2 :

Find the sum of the series from 1 upto the preceding term of the first term in the given series.

(1 + 2 + 3 +........+ 14)

Sum of natural numbers  =  n(n + 1) / 2

n  =  14

=  14(14 + 1) / 2

=  7 ⋅ 15

=  105  -----(2)

Step 3 :

(1)  =  (2)

Required sum  =  3240 - 105

= 3135

Example 3 :

Find the sum of

1 + 4 + 9 + ............. + 1600

Solution :

By observing the given series, we find that every term can be represented in the form of square.

1 + 4 + 9 + ............. + 1600  =  12 + 22 + 32 + ............. + 402

Sum of squares  =  n(n+ 1) (2n + 1) / 6

n  =  40

=  40 (40 + 1) (2⋅40 + 1) / 6

=  (40 ⋅ 41 ⋅ 81) / 6

=  22140

Example 4 :

Find the sum of the following series

113 + 12 3 + ............... +k3 where k = 50

Solution :

By applying the value of k, we get

(113 + 123 + ..........+503)

Step 1 :

Find the sum of the series from 1 upto the last term 50.

(13 + 23 + ..........+ 503)

Sum of cubes  =  [n(n + 1)/2]2

n  =  50

=  [50(50 + 1)/2]2

=  [25 ⋅ 51]2

=  12752

=  1625625  ----(1)

Step 2 :

Find the sum of the series from 1 upto the preceding term of the first term in the given series.

(13 + 23 + ..........+ 103)

=  [10(10 + 1)/2]2

=  (5 ⋅ 11)2

=  552

=  3025  ----(2)

Step 3 :

(1)  =  (2)

Required result  = 1625625 - 3025

=  1622600 Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us :

v4formath@gmail.com

You can also visit the following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6

1. Click on the HTML link code below.

Featured Categories

Math Word Problems

SAT Math Worksheet

P-SAT Preparation

Math Calculators

Quantitative Aptitude

Transformations

Algebraic Identities

Trig. Identities

SOHCAHTOA

Multiplication Tricks

PEMDAS Rule

Types of Angles

Aptitude Test 