**Arithmetic Progression (A.P. or AP)**

Let us recall some formulae and properties studied earlier.

A sequence *a*_{1}, *a*_{2}, *a*_{3}, …, *a _{n}* is called arithmetic sequence or arithmetic progression if

*a*

_{(n+1)}=

*a*+

_{n}*d*,

*n*∈N, where

*a*

_{1}is called the first term and the constant term

*d*is called the common difference of the A.P.

Let us consider an A.P. (in its standard form) with first term *a* and common difference *d*, i.e., *a*, *a*+*d*, *a*+2*d*, ….

Then the *n*th term (general term) of the A.P. is *a _{n}*=

*a*+(

*n*-1)

*d*.

We can verify the following simple properties of an A.P.:

(i) If a constant is added to each term of an A.P., the resulting sequence is also an A.P.

(ii) If a constant is subtracted from each term of an A.P., the resulting sequence is also an A.P.

(iii) If each term of an A.P. is multiplied by a constant, then the resulting sequence is also an A.P.

(iv) If each term of an A.P. is divided by a non-zero constant then the resulting sequence is also an A.P.

Here, we shall use the following notations for an arithmetic progression:

*a*= the first term, *ℓ*= the last term, *d*= common difference, *n*= the number of terms. *S _{n}*= the sum to

*n*terms of A.P.

Let

*a*,

*a*+

*d*,

*a*+2

*d*,

*a*+(

*n*-1)

*d*be an A.P. Then

*ℓ*=

*a*+(

*n*-1)

*d*

*S*=½

_{n}*n*[2

*a*+(

*n*-1)

*d*]

We can also write,

*S*=½

_{n}*n*(

*a*+

*ℓ*)

**Formula for Sum to n Terms Of an AP**

Let

*S*=

_{n}*a*+(

*a*+

*d*)+(

*a*+2

*d*)+…+(

*a*+(

*n*-2)

*d*)+(

*a*+(

*n*-1)

*d*) … (1)

Writing the expression in the reverse order, we get

*S*=(

_{n}*a*+(

*n*-1)

*d*)+(

*a*+(

*n*-2)

*d*)+…+(

*a*+2

*d*)+(

*a*+

*d*)+

*a*… (2)

Adding (1) and (2) Vertically, we get

*S*=[2

_{n}*a*+(

*n*-1)

*d*]+[2

*a*+(

*n*-1)

*d*]+…+[2

*a*+(

*n*-1)

*d*] … (

*n*expressions)

⇒

*S*=½

_{n}*n*[2

*a*+(

*n*-1)

*d*]

Alternative form for the Sum Formula

*S*=½

_{n}*n*[

*a*+

*a*+(

*n*-1)

*d*]

⇒

*S*=½

_{n}*n*[

*a*+

*ℓ*]

where

*ℓ*=

*a*+(

*n*-1)

*d*is the last term of the AP

(Proof 1). The first and last terms of an AP are *a* and *ℓ* respectively. Show that the sum of the *n*th term from the beginning and the *n*th term form the end is (*a*+*ℓ*).

Solution:

In the given AP, first term =*a* and last term =*ℓ*.

Let the common difference be *d*.

Then, *n*th term from the beginning is given by

*a*=

_{n}*a*+(

*n*-1)

*d*… (1)

Similarly,

*n*th term from the end is given by

*a*=

_{n}*ℓ*-(

*n*-1)

*d*… (2)

Adding (1) and (2), we get

*a*+(

*n*-1)

*d*+{

*ℓ*-(

*n*-1)

*d*}

=

*a*+(

*n*-1)

*d*+

*ℓ*-(

*n*-1)

*d*

=

*a*+

*ℓ*

Hence, the sum of the

*n*th term from the beginning and the

*n*th term from the end (

*a*+

*ℓ*).

(Proof 2). If the *p*th term of an AP is *q* and its *q*th term is *p* then show that its (*p*+*q*)th term is zero.

Solution:

In the given AP, let the first term be *a* and the common difference be *d*.

Then, *a _{n}*=

*a*+(

*n*-1)

*d*

*a*=

_{p}*a*+(

*p*-1)

*d*=

*q*… (i)

*a*=

_{q}*a*+(

*q*-1)

*d*=

*p*… (ii)

On subtracting (i) from (ii), we get

*q*–

*p*)

*d*=(

*p*–

*q*)

*d*=-1

Putting

*d*=-1 in (i), we get

*a*=(

*p+q*-1)

Thus,

*a*=(

*p+q*-1) and

*d*=-1 Now,

*a*

_{(p+q)}=

*a*+(

*p+q*-1)

*d*

*p+q*-1)+(

*p+q*-1)(-1)

=(p+q-1)-(

*p+q*-1)=0

Hence, the (

*p+q*)th term is 0 (zero).

Example 1. Find the sum of each of the following arithmetic series:

(i) 7+10½+14+⋯ +84

(ii) 34+32+30+⋯ +10

(iii) (-5)+(-8)+(-11)+⋯ +(-230)

Solution:

(i) The given arithmetic series is 7+10½+14+⋯ +84.

Here, *a*=7, *d*=10½-7=3½ and *ℓ*=84.

Let the given series contains *n* terms. Then, *a _{n}*=84.

*a*=

_{n}*a*+(

*n*-1)

*d*]

7+(

*n*-1)∙3½=84 … (×2)

14+(

*n*-1)∙7=168

*n*-1=(168-14)÷7

*n*-1=22

*n*=23

*S _{n}*=½

*n*[

*a*+

*ℓ*]

∴ Required sum

*S*

_{23}=½∙23∙(7+84)

=½∙2030

=1046½

(ii) The given arithmetic series is 34+32+30+⋯ +10.

Here, *a*=34, *d*=32-34=-2 and *ℓ*=10.

Let the given series contain *n* terms. Then, *a _{n}*=10.

*a*=

_{n}*a*+(

*n*-1)

*d*]

34+(

*n*-1)⋅(-2)=10

-2

*n*+36=10

-2

*n*=10-36=-26

*n*=13.

*S _{n}*=½

*n*[

*a*+

*ℓ*]

∴ Required sum

*S*

_{13}=½∙13∙(34+10)

=½∙286

(iii) The given arithmetic series is (-5)+(-8)+(-11)+⋯ +(-230).

Here, *a*=-5, *d*=-8-(-5)=-8+5=-3 and *ℓ*=230.

Let the given series contain *n* terms. Then, *a _{n}*=-230.

*a*=

_{n}*a*+(

*n*-1)

*d*]

-5+(

*n*-1)⋅(-3)=-230

-3

*n*-2=-230

-3

*n*=-230+2=-228

*n*=76

*S _{n}*=½

*n*[

*a*+

*ℓ*]

∴ Required sum

*S*

_{76}=½∙76∙(-5-230)

=-8930

**Find the sum of each arithmetic series Ex2 to Ex11.**

Ex2. 4+8+12+…+200

Solution:

12-8=4

The common difference is 4.

*a*

_{1}=4,

*a*=200,

_{n}*d*=4

*a*=

_{n}*a*

_{1}+(

*n*-1)

*d*

200=4+(

*n*-1)4

4

*n*=200

*n*=50

Find the sum of the series.

*S*=½⋅

_{n}*n*⋅(

*a*

_{1}+

*a*)

_{n}*S*

_{50}=½⋅50⋅(4+200)

=5100

Answer:

5100

Ex3. -18+(-15)+(-12)+…+66

Solution:

-12-(-15)=3

The common difference is 3.

*a*

_{1}=-18,

*a*=66

_{n}Find the value of

*n*.

*a*=

_{n}*a*

_{1}+(

*n*-1)

*d*

66=-18+(

*n*-1)3

3

*n*=87

*n*=29

Find the sum.

*S*=½

_{n}*n*(

*a*

_{1}+

*a*)

_{n}*S*

_{29}=½⋅29⋅(-18+66)

=696

Answer: 696

Ex4. -24+(-18)+(-12)+…+72

Solution:

-12-(-18)=6

The common difference is 6.

*a*

_{1}=-24,

*a*=72

_{n}Find the value of

*n*.

*a*=

_{n}*a*

_{1}+(

*n*-1)

*d*

72=-24+(

*n*-1)6

6

*n*=102

*n*=17

Find the sum.

*S*=½

_{n}*n*(

*a*

_{1}+

*a*)

_{n}*S*

_{17}=½⋅17⋅(-24+72)

=408

Answer: 408

Ex5. *a*_{1}=12, *a _{n}*=188,

*d*=4

Solution:

*a*=

_{n}*a*

_{1}+(

*n*-1)

*d*

188=12+(

*n*-1)4

4

*n*=180

*n*=45

Find the sum of the series.

*S*=½⋅

_{n}*n*⋅(

*a*

_{1}+

*a*)

_{n}*S*

_{45}=½⋅45⋅(12+188)

=4500

Answer:

4500

Ex6. *a _{n}*=145,

*d*=5,

*n*=21

Solution:

*a*=

_{n}*a*

_{1}+(

*n*-1)

*d*

145=

*a*

_{1}+(21-1)5

*a*

_{1}=45

Find the sum of the series.

*S*=½⋅

_{n}*n*⋅(

*a*

_{1}+

*a*)

_{n}*S*

_{21}=½⋅21⋅(45+145)

=1995

Answer:

1995

Ex7. the first 50 natural numbers

Solution:

*S*

_{50}=½⋅50⋅(1+50)

=½⋅50⋅51

=1275

Answer: 1275

Ex8. the first 100 odd natural numbers

Solution: Here *a*_{1}=1, *a*_{100}=199 and *n*=100

Find the sum.

*S*=½

_{n}*n*(

*a*

_{1}+

*a*)

_{n}*S*

_{100}=½⋅100⋅(1+199)

=10,000

Answer: 10,000

Ex9. the first 200 odd natural numbers

Solution:

Here *a*_{1}=1 and *a*_{200}=399.

Find the sum.

*S*=½

_{n}*n*(

*a*

_{1}+

*a*)

_{n}*S*

_{200}=½⋅200⋅(1+399)

=40,000

Answer: 40,000

Ex10. the first 100 even natural numbers

Solution: Here *a*_{1}=2 and *a*_{100}=200.

Find the sum.

*S*=½

_{n}*n*(

*a*

_{1}+

*a*)

_{n}*S*

_{100}=½⋅100⋅(2+200)

=10,100

Answer: 10,100

Ex11. the first 300 even natural numbers

Solution:

Here *a*_{1}=2, *a*_{300}=300 and *n*=300.

Find the sum.

*S*=½

_{n}*n*(

*a*

_{1}+

*a*)

_{n}*S*

_{300}=½⋅300⋅(2+600)

=90,300

Answer: 90,300

Ex12. Determine the sum of the series: 19+22+25+…+121

solution:

*a*=19 and

*d*=3

*a*=3

_{n}*n*+16=121=

*ℓ*

3

*n*=105

*n*=35

*S*=½

_{n}*n*(

*a*+

*ℓ*)

*S*

_{35}=½⋅35⋅(19+121)=35⋅½⋅140=35⋅70=2450

Ex13. Find the sum of the series 1+3.5+6+8.5+…+101.

Solution:

This is an arithmetic series, because the difference between the terms is a constant value, 2.5. We also know that the first term is 1, and the last term is 101. But we do not know how many terms are in the series. So we will need to use the formula for the last term of an arithmetic progression,

*ℓ*=

*a*+(

*n*-1)

*d*

to give us 101=1+(

*n*-1)×2.5.

Now this is just an equation for

*n*, the number of terms in the series, and we can solve it. If we subtract 1 from each side we get

*n*-1)×2.5

and then dividing both sides by 2.5 gives us

*n*-1

so that

*n*=41. Now we can use the formula for the sum of an arithmetic progression, in the version using

*ℓ*, to give us

*S*=½

_{n}*n*(

*a*+

*ℓ*)

*S*

_{41}=½×41×(1+101)

=½×41×102

=41×51

2091

Ex14. Find the sum of -6+1+8+15+…+141.

solution:

The series is arithmetic with *u*_{1}=-6, *d*=7 and *u _{n}*=141. First we need to find

*n*.

*u*

_{1}+(

*n*-1)

*d*=141

-6+7(

*n*-1)=141

7(

*n*-1)=147

*n*-1=21

*n*=22

Using

*S*=½

_{n}*n*(

*u*

_{1}+

*u*)

_{n}*S*

_{22}=½⋅22⋅(-6+141)

=11⋅135=1485

Ex15. Find the number of terms of the AP -12, -9, -6, …, 21.

If 1 is added to each term of this AP then the sum of all terms of the AP thus obtained.

Solution:

The given AP is -12, -9, -6, …, 21.

Here, *a*=-12, *d*=-9-(-12)=-9+12=3 and *ℓ*=21.

Suppose there are *n* terms in the AP.

*ℓ*=

*a*=21

_{n}[

*a*=

_{n}*a*+(

*n*-1)

*d*]

-12+(*n*-1)⋅3=21

3*n*-15=21

3*n*=21+15=36

*n*=12.

Thus, there are 12 terms in the AP.

If 1 is added to each term of the AP, then the new AP so obtained is -11, -8, -5, …, 22.

Here, first term,

*a*=-11, last term,

*ℓ*=22 and

*n*=12.

Get the sum of the terrns of this AP.

*S*=½

_{n}*n*(

*a*+

*ℓ*)

∴

*S*

_{12}=½⋅12⋅(-11+22)

=6⋅11=66

Hence, the required sum is 66.

Ex16. The sum of the first 20 odd natural numbers is

(a) 100 (b) 210 (c) 400 (d) 420

Solution:

The first 20 odd natural numbers are 1, 3, 5, …, 39.

These numbers are in AP.

Here *a*=1, *ℓ*=39 and *n*=20.

∴ Sum of first 20 odd tural number

*S*=½

_{n}*n*(

*a*+

*ℓ*)

*S*

_{20}=½⋅20⋅(1+19)

=10⋅40

=400.

Answer: (c) 400

Ex17. Find the sum of all odd numbers between 0 and 50.

Solution:

All odd numbers between 0 and 50 are 1, 3, 5, 7, …, 49.

This is an AP in which *a*=1, *d*=(3-1)=2 and *ℓ*=49.

Let the number of terms be *n*.

Then, *a _{n}*=49

*a*+(

*n*-1)

*d*=49

1+(

*n*-1)⋅2=49

2

*n*=50

*n*=25

∴ Required sum =½

*n*(

*a*+

*ℓ*)

=25⋅½⋅50

=25⋅25=625

Hence, the required sum is 625.

Ex18: Finding the Sum of an Arithmetic Series

Find the sum of all the odd numbers between 51 and 99, inclusive.

Solution:

First, use *a*_{1}=51, *a _{n}*=99 to find

*n*:

*a*=

_{n}*a*

_{1}+(

*n*-1)

*d*

99=51+(

*n*-1)2

*n*=25

Now, find

*S*

_{25}.

*S*=½

_{n}*n*(

*a*

_{1}+

*a*)

_{n}*S*

_{25}=½⋅25⋅(51+99)

=1,875

Ex19. Find the sum of all integers between 100 and 1000 which are divisible by 9.

Solution: The first integer greater than 100 and divisible by 9 is 108 and the integer just smaller than 1000 and divisible by 9 is 999. Thus, we have to find the sum of the series.

Here

*t*=

*a*=108,

*d*=9 and

*ℓ*=999

Let

*n*be the total number of terms in the series be

*n*. Then

*n*-1) … (÷9)

111=12+(

*n*-1)

*n*=100

Hence, the required sum

*S*=½

_{n}*n*(

*a*+

*ℓ*)

= ½⋅100⋅(108+999)

=50(1107)=55350.

Lets read post Numbers which are divisible by or multiples of an integer in Arithmetic Sequence

Question 1. (5+13+21+…+181)=?

(a) 2476 (b) 2337 (c) 2219 (d) 2139

Solution:

Here,

*a*=5,

*d*=(13-5)=8 and

*ℓ*=181.

Let the number of terms be

*n*. Then

*a*=181

_{n}*a*+(

*n*-1)

*d*=181

5+(

*n*-1)⋅8=181

8

*n*=184

*n*=23

∴ Required sum =½

*n*(

*a*+

*ℓ*)

Hence, the required sum is 2139.

Answer: (d) 2139

Q2. A local theater has 30 seats in the first row and 50 rows in all. Each successive row contains two additional seats. How many seats are in the theater?

solution:

I need to know how many seats are in the 50th row. I will find a formula for an to get started.

*a*=2

_{n}*n*+

*x*

*a*

_{1}=2(1)+

*x*

30=2+

*x*

28=

*x*

formula for sequence:

*a*=2

_{n}*n*+28.

Computation of number of seats in 50th row:

*a*

_{50}=2(50)+28=128.

I need to add these numbers to get my answer: 30+32+34+…..+128.

I will use the formula

*S*=½(

_{n}*a*

_{1}+

*a*)

_{n}*S*

_{50}=½⋅50⋅(30+128)=25(158)=3950

Answer: There are 3950 seats.

Q3. An outdoor amphitheater has 40 seats in the first row, 41 seats in the second row, 42 seats in the third row. The pattern continues. The amphitheater has 70 rows.

How many seats does the amphitheater contain?

solution:

I need to know how many seats are in the 70th row. I will find a formula for an to get started.

*a*=1

_{n}*n*+

*x*

*a*

_{1}=1(1)+

*x*

40=1+

*x*

39=

*x*

formula for sequence:

*a*=

_{n}*n*+39.

Computation of number of seats in 70th row:

*a*

_{70}=70+39=109.

I need to add these numbers to get my answer: 40+41+42+…..+109.

I will use the formula

*S*=½(

_{n}*a*

_{1}+

*a*).

_{n}*S*

_{70}=½⋅70⋅(40+109)=35(149)=5215

Answer: There are 5215 seats

Q4. The first and the last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?

Solution:

Suppose there are *n* terms in the AP.

Here, *a*=17, *d*=9 and *ℓ*=350.

*a*=350

_{n}[

*a*=

_{n}*a*+(

*n*-1)

*d*]

17+(

*n*-1)⋅9=350

9

*n*+8=350

9

*n*=350-8=342

*n*=38.

Thus, there are 38 terms in the P.

*S*=½

_{n}*n*(

*a*+

*ℓ*)

*S*

_{38}=½⋅38⋅(17+350)

=19⋅367

=6973

Hence, the required sum is 6973.

Q5. (Contests) The prizes in a weekly radio contest began at $150 and increased by $50 for each week that the contest lasted. If the contest lasted for eleven weeks, how much was awarded in total?

Solution:

Given, *a*_{1}=150, *d*=50 and *n*=11.

Find the value of *a*_{11}.

*a*=

_{n}*a*

_{1}+(

*n*-1)

*d*

*a*

_{11}=

*a*

_{1}+(11-1)

*d*

=150+10⋅50

=650

Find the sum.

*S*=½⋅

_{n}*n*⋅(

*a*

_{1}+

*a*)

_{n}=11⋅½⋅(150+650)

=4400

A cash prizes totaled $4400 for the eleven week contest.

Answer: $4400

Q6. (Drama) Laura has a drama performance in 12 days. She plans to practice her lines each night. On the first night she rehearses her lines 2 times. The next night she rehearses her lines 4 times. The third night she rehearses her lines 6 times. On the eleventh night, how many times has she rehearsed her lines?

Solution:

The arithmetic sequence that represents the situation is 2, 4, 6, ….

Substitute 2 for *a*_{1}, 2 for *d*, and 11 for *n* in the formula for the *n*th term and find *a*_{11}.

*a*=

_{n}*a*

_{1}+(

*n*-1)

*d*

*a*

_{11}=2+(11-1)2

=2+20

=22

Substitute 2 for

*a*

_{1}, 22 for

*a*, 11 for

_{n}*n*in the Sum formula

*S*=½

_{n}*n*(

*a*

_{1}+

*a*)

_{n}=11⋅½⋅(2+22)

=11⋅12

=132

Answer: 132

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