Finding a Formula for Finding a Formula for Sum of a SequenceSum of a Sequence

KAPIL VERMA10TH ‘A’

21

A sequence is arithmetic if

each term – the previous term = d

where d is a constante.g. For the

sequence ...,8,6,4,2d = 2nd term – 1st

term= 3rd term – 2nd term . . . = 2

Arithmetic Sequence

The 1st term of an arithmetic sequence is

given the letter a.

Arithmetic Sequence

An arithmetic sequence is of the form...,3,2,, dadadaa

Notice that the 4th term has 3d added so, for example, the 20th term will be

da 19

The nth term of an Arithmetic Sequence is

dnatn )1(

Arithmetic Series

When the terms of a sequence are added we get a seriese.g. The sequence

gives the series

...,8,6,4,2...8642

The Sum of an Arithmetic SeriesWe can derive a formula that can be used for

finding the sum of the terms of an arithmetic series

Arithmetic Series

...4321

e.g. Find the sum of the 1st 10 terms of the series

Solution: Writing out all 10 terms we have

10987654321

Adding the 1st and last terms gives 11. Adding the 2nd and next to last terms

gives 11. The 10 terms give 5 pairs of size 11 ( =

55 ).Writing this as a formula we have

)(2

lan

la

where l is the last

term

With an odd number of terms, we can’t pair up all the terms. e.g.

Arithmetic Series

7654321 However, still works since we can miss

out the middle term

)(2

lan

giving n = 6.

Now we add the middle term

)71(2

6)(

2 la

nWe get

However, still works since we can miss

out the middle term

With an odd number of terms, we can’t pair up all the terms. e.g.

Arithmetic Series

7654321 )(

2la

n

Together we have which is )71(2

7 )(

2la

n

giving n = 6.

Now we add the middle term

which equals )71(2

1

4

)71(2

6)(

2 la

nWe get

)(2

lan

Sn

For any arithmetic series, the sum of n terms is given by

Substituting for l in the formula for the

sum gives an alternative form:

lSince the last term is also the nth term,

))1(2(2

dnan

Sn

dna )1(

SUMMARY

)(2

lan

Sn

The sum of n terms of an arithmetic series

is given by

...,3,2,, dadadaa

An arithmetic sequence is of the form

The nth term is dnatn )1(

or

))1(2(2

dnan

Sn

e.g.1 Find the 20th term and the sum of 20 terms of the series:

2 + 5 + 8 + 11 + 14 + 17 + . . . Solution: The series is arithmetic.

203,2 nda and

20u 59)3(192 dnatn )1(

5920 ulwhere

)592(2

2020S 610

)(2

lan

Sn Either

or ))1(2(

2dna

nSn 610)3)19(4(

2

2020 S

e.g.2 The common difference of an arithmetic series is -3 and the sum of the first 30 terms is 255. Find the 1st term.

Solution:

255303 30S and , nd

))1(2(2

dnan

S n

))3(292(2

30255 a

)872(15255 a

87215

255 a

52 aa28717

Exercises

1. The 1st term of an A.P. is 20 and the sum of 16 terms is 280. Find the last term and the common difference.

2. 10

1

104n

Solution: )(

2la

nS n )20(8280 l l 15

d152015 dnaltn )1( 3

1 d

)(3010)10(410 10 lun

120)306(510 S

Find the sum of the series given by

)(2

lan

Sn

We can see the series is arithmetic so,

Substituting n = 1, 2 and 3, we get 6, 2, 2