**Arithmetic**** progressions****:**** Basic**** concepts**

**Sequence: **Some numbers arranged in a definite order, according to a definite rule, are said to form a sequence.

ex (I) 1, 2, 3,………

(II) 100, 70, 40, 10 ………

(III) -1·0, -1·5, -2·0, -2·5, …….

**Arithmatic progressions (AP):**

An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

→ this fixed number is called the common difference of A.P.

→ common differences can be positive, negative or zero

** Exercise – 5·1**

**1. In which of the following situations, does the list of numbers involved make an arithmetic progression, any why?**

**(I) the taxi fare after each km when the fare is rupees 15 for the first km and rupees 8 for each additional km.**

**Solution:**

yes, it is in A.P.

15, 23, 31, ……… each succeeding term is obtained by adding 8 in its preceding.

**(II) the amount of air present in a cylinder when a vacuum pump removes 1/4 of the additional km.**

**Solution:**

No. volume: v, 3/4v, (3/4)² …….

**(III) the cost of digging a well after every metre of digging, when it costs rupees 150 for the first metre and rises by rupees 50 for each subsequent metre. **

**Solution:**

yes, 150, 200, 250 —– succeeding term is obtained by adding 50 in its preceding.

**(IV) the amount of money in the account every year, when rupees 10000 is deposited at compound interest at 8% per annum.**

**Solutions:**

**2. Write the first four terms of the AP, when the first term a and the common difference d are given as follows ÷**

**(I) a=10 , d=10**

**Solution:**

a_{2 }= a+d = 10+10 =20

a_{3 }= a_{2}+d = 20+10 = 30

a_{4 }= a_{3}+d = 30+10 = 40

a_{5 }= a_{4}+d = 40+10 = 50

**(II) a=-2,d=0**

**Solution:**

a_{2 }= a+d = -2+0 =-2

a_{3 }= a_{2}+d = -2+0 = -2

a_{4 }= a_{3}+d = -2+0 =-2

a_{5 }= a_{4}+d = -2+0 =-2

**(III) a=4 , d=-3**

**Solution:**

a_{2 }= a+d = 4-3 = 1

a_{3 }= a_{2}+d =1-3 = -2

a_{4 }= a_{3}+d =-2-3 =-5

a_{5 }= a_{4}+d = -5-3 =-8

**(IV) a=-1 , d=1/2**

**Solution:**

a_{2 }= a+d = -1+1/2 = -2+1/2=-1/2

a_{3 }= a_{2}+d =-1/2+1/2=0

a_{4 }= a_{3}+d =0+1/2=1/2

a_{5 }= a_{4}+d =1/2+1/2 = 2/2=1

**(V) a=-1·25 , d=-0·25**

**Solution:**

a_{2 }= a+d =-1·25+(-0.25)= -1·25-0·25=-1·50

a_{3 }= a_{2}+d =-1·50+(-0·25) = -1·50-0·25=-1·75

a_{4 }= a_{3}+d =-1·75+(-0·25) = -1·75-0·25=-2·00

a_{5 }= a_{4}+d =-2·00+(-0·25) = -2·00-0·25=-2·25

**Q.3 For the following APs, write the first term and the common difference ÷**

**(I) 3,1,-1,-3, …..**

**Solution:**

first term =a=3

common difference =1-3 = -2

**(II) -5,-1,3,7, …..**

**Solution:**

first term =-5

common difference = -1-5 = -6

**(III) 1/3,5/3,9/3,13/3, …..**

**Solution:**

first term = 1/3

common difference = 5/3-1/3=4/3

**(IV) 0·6,1·7,2·8,3·9, …..**

**Solution:**

first term =a = 0·6

common difference = 1·7-0·6 =1·1

**Q.4 Which of the following are APS? if they form an A.P., find the common difference d write three more terms.**

**(I) 2,4,8,16,……..**

**Solution:**

first term (a) = 2

d_{1}=4-2=2

d_{2}=8-4=4

d_{3}=16-8=8

d_{1}≠d_{2}≠d_{3}

No above sequence is not in A.P.

**(II) 2,5/2,3,7/2,…..**

**Solution:**

first term a=2

d_{1}=5/2-2=5-4/2=1/2

d_{2}=3-5/2=6-5/2=1/2

d_{3}=7/2-3=7-6/2=1/2

d=d_{1}=d_{2}=d_{3}

common difference d=-1/2

yes, it is in A.P.

a_{5}=a_{4}+d=7/2+1/2=8/2=4

a_{6}=a_{5}+d=4+1/2=8+1/2=9/2

a_{7}=a_{6}+d=9/2+1/2=10/2=5

a_{8}=a_{7}+d=5+1/2=10+1/2=11/2

Hence next three terms = 4,9/2,5,11/2,……

**(III) -1·2,-3·2,-5·2,-7·2,…….**

**Solution:**

d_{1}=-3·2-(-1·2)=-3·2+1·2=-2·0

d_{2}=-5·2-(-3·2)=-5·2+3·2=-5·2+3·2=-2·0

d_{3}=-7·2-(-5·2)=-7·2+5·2=-2·0

here d=d_{1}=d_{2}=d_{3}=-2·0

Hence yes above sequence form an A.P. next three terms

a_{5}=a_{4}+d=-7·2+(-2·0)=-7·2-2·0=-9·2

a_{6}=a_{5}+d=-9·2+(-2·0)=-9·2-2·0 =-11·2

a_{7}=a_{6}+d=-11·2+(-2·0)=-11·2-2·0=-13·2

**(IV) -10,-6,-2,2…….**

d_{1}=-6-(-10)=-6+10=4

d_{2}=-2-(-6)=-2+6=4

d_{3}=2-(-2)=2+2=4

d=d_{1}=d_{2}=d_{3}=4

yes, it is in A.P.

common difference =4

next three terms.

a_{5}=a_{4}+d=2+4=6

a_{6}=a_{5}+d=6+2=8

a_{7}=a_{6}+d=8+2=10

**(V) 3,3+√2,3+2√2,3+3√2,……**

d_{1}=3+√2 – 3 = √2

d_{2}=3+2√2 – (3+√2)= 3+2√2 – 3 -√2 = √2

d_{3}=3+3√2 – (3+2√2) = 3+3√2 – 3 -2√2 = √2

d=d_{1}=d_{2}=d_{3}=√2

Yes above sequence form an A.P. and the next three terms are:

a_{5}=a_{4}+d= 3+3√2+√2 = 3+4√2

a_{6 }= a_{5}+d= 3+4√2+√2 = 3+5√2

a_{7}=a_{6}+d= 3+5√2 + √2 = 3+6√2

**(VI) 0.2,0.22,0.222,0.2222,…….**

d_{1}= 0.22 – 0.2= 0.20

d_{2}= 0.222 – 0.22= 0.002

d_{1}≠d_{2}

No above sequence is not in A.P.

**(VII) 0,-4,-8,-12,…..**

d_{1}=-4 – 0 = -4

d_{2}= -8-(-4) = -8 + 4=-4

d_{3}=-12-(-8)=-12+8=-4

d=d_{1}=d_{2}=d_{3}=-4

Yes above sequence form an A.P. and the next three terms are:

a_{5}=a_{4}+d= -12+(-4)=-12-4 = -16

a_{6 }= a_{5}+d= -16+(-4)= -16-4 =-20

a_{7}=a_{6}+d= -20+(-4)=-20-4=-24

**(VIII) -1/2,-1/2,-1/2,-1/2**

d_{1}=-1/2 – (-1/2)=-1/2+1/2=0

d_{2}= -1/2 – (-1/2)=-1/2+1/2=0

d_{3}=-1/2 – (-1/2)=-1/2+1/2=0

d=d_{1}=d_{2}=d_{3}=0

Yes above sequence form an A.P. and the next three terms are:

a_{5}=a_{4}+d= -1/2+0=-1/2

a_{6 }= a_{5}+d= -1/2+0=-1/2

a_{7}=a_{6}+d= -1/2+0=-1/2

**(IX) 1,3,9,27,…….**

d_{1}=3-1=2

d_{2}=9-3=6

d_{1}≠d_{2}

No above sequence is not in A.P.

**(X) a,2a,3a,4a,…….**

d_{1}=2a-a=a

d_{2}=3a – 2a=a

d_{3}=4a-3a=a

d=d_{1}=d_{2}=d_{3}=a

Yes above sequence form an A.P. and the next three terms are:

a_{5}=a_{4}+d= 4a+a=5a

a_{6 }= a_{5}+d=5a+a=6a

a_{7}=a_{6}+d=6a+a=7a

**(XI) a,a²,a³,a ^{4}**,……

d_{1}=a² – a =a(a-1)

d_{2}=a³ – a² =a²(a-1)

d_{1}≠d_{2}

No above sequence is not in A.P.

**(XII)√2,√8,√18,√32,…**

d_{1}=√8-√2=2√2-√2=√2

d_{2}=√18-√8=√(3×3×2)-√(2×2×2)=3√2-2√2=√2

d_{3}=√32-√18=√(4×4×2)-3√2=4√2-3√2=√2

d=d_{1}=d_{2}=d_{3}=√2

Yes above sequence form an A.P. and the next three terms are:

a_{5}=a_{4}+d=√32+√2=√64

a_{6 }= a_{5}+d=√64+√2=√128

a_{7}=a_{6}+d=√128+√2=√256

**(XIII)√3,√6,√9,√12,….**

d_{1}=√6-√3

d_{2}=√9-√6

d_{1}≠d_{2}

No above sequence is not in A.P.

**(XIV) 1²,3²,5²,7²,…..**

d_{1}=3²- 1²=9-1=8

d_{2}=5²-3²=25-9=16

d_{3}=7²- 5²=49-25=24

d_{1}≠d_{2}≠d_{3}

No above sequence is not in A.P.

**(XV) 1²,3²,5²,73,…..**

d_{1}=3²- 1²=9-1=8

d_{2}=5²-3²=25-9=16

d_{3}=5²- 73=25-73=-48

d_{1}≠d_{2}≠d_{3}

No above sequence is not in A.P