LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

INTRODUCTION & SEQUENCE

**ARITHMETIC PROGRESSION (A.P) –
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**Sequence –
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**Some numbers arranged in a definite order according to some definite rule are said to form a sequence. The number occurring at the nth place of a sequence is called its nth term, denoted by Tₑ or aₑ.
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**Example.) Consider the rule, Tₑ= (3e + 5)
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**Putting, e = 1, 2, 3, 4, 5, 6, 7,……………., we get
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**T₁ = (3e + 5) = 8
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**T₂ = (3e + 5) = 11
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**T₃ = (3e + 5) = 14, and so on ……..
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**Thus the numbers 8, 11, 14, 17, 20, 23,………… form a sequence. In this sequence, the first term is 8, the second term is 11, the third term is 14, and so on. The eth term is (3e + 5). The eth term in a sequence is called its general term.**

**Arithmetic Progression (A.P) –
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**A sequence, in which each term differs from its preceding term by a constant, is called an arithmetic progression which can be written as A.P. The constant difference is called the common difference.
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**Example.1) Show that the progression 7, 12, 17, 22, 27,………..is in A.P. Write its first term and common difference.
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**Ans.) The given progression is 7, 12, 17, 22, 27,………..
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**We have, (12 – 7) = 5
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** (17 – 12) = 5
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** (22 – 17) = 5
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** (27 – 22) = 5
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**So, (12 – 7) = (17 – 12) = (22 – 17) = (27 – 22) = 5 (constant)
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**So, the given progression is an A.P
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**It’s first term = 7, and common difference = 5 (Ans.)**

**Example.2) Show that the progression 7, 4, 1, - 2, - 5, - 8,……….. is n A.P. Write its first term and the common difference
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**Ans.) The given progression is 7, 4, 1, - 2, - 5, - 8,………..
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**We have, (4 – 7) = - 3,
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** (1 – 4) = - 3,
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** (- 2 – 1) = - 3,
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** {- 5 – (- 2)} = (- 5 + 2) = - 3
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** {- 8 – (- 5)} = (- 8 + 5) = - 3
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**So, (4 – 7) = (1 – 4) = (- 2 – 1) = {- 5 – (- 2)} = {- 8 – (- 5)} = - 3 (constant)
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**So, the given progression is an A.P.
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**Its first term = 7, and common difference = - 3. (Ans.)**

**Example.3) Show that each of the following progressions is an A.P. Find the common difference and the next term of each –
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**(i) √7, √28, √63,………… **

**Ans.) The given terms are √7, √28, √63,…………
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**We write them as √7, √(4 X 7), √(9 X 7),…………
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** => √7, √(2² X 7), √(3² X 7),…………
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** => √7, 2√7, 3√7,…………
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** We have, (2√7 - √7) = √7
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** (3√7 - 2√7) = √7
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**So, (2√7 - √7) = (3√7 - 2√7) = √7 (constant)
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**So the given progression is an A.P. in which the common difference is √7. Clearly the next term is 4√7 = √(4² X 7) = √(16 X 7) = √112 (Ans.)
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**(ii) √18, √50, √98,…………….**

**The given terms are, √18, √50, √98,…………….
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**We may write as, √(9 X 2), √(25 X 2), √(49 X 2),…………….
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** => √(3² X 2), √(5² X 2), √(7² X 2),…………..
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** => 3√2, 5√2, 7√2,……………
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**We have, (5√2 - 3√2) = 2√2
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** (7√2 - 5√2) = 2√2
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**So, (5√2 - 3√2) = (7√2 - 5√2) = 2√2 (constant)
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**So the given progression is an A.P. in which the common difference is 2√2. Clearly the next term is 9√2 = √(9² X 2) = √(81 X 2) = √162 (Ans.)**

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