Sequence and Series Class 11 Notes

Sequence and Series Class 11 Notes
Sequence :Â

A set of numbers arranged in a definite order according to some rule is called a sequence.

Join the Group :

OR

A sequence is a function of natural numbers with codomain is the set of Real Numbers (Complex Numbers) . If Range is a subset of Real Numers (Complex Numbers) then it is called a real sequence (Complex Sequence).

OR

A mapping f : N â†’ A then f (n) = tn , n âˆˆ N is called a sequence to be denoted byÂ

Range < fn > = { f (1) , f (2) , f (3) , ———- } = { t1 , t2 , t3 , ——- } OR < fn >Â  orÂ  { tn }Â

OR

A sequence is a function whose domain is the of natural numbers (N) or some subsets of the type {1 , 2 , 3 , 4 , 5 , ——– , k} .Â

Example : 2 , 4, 6 , 8 ,—— is a sequence .

Note : Sequence is said to be finite or infinite sequence according it has finite or infinite number of terms.

Series :Â

IfÂ  a1 , a2 , ——– , an , ——– is a sequence , then the expression a1 + a2 + —— + an + —— is called the series . The series is said to be finite or infinite according as the given series is finite or infinite.

Example :Â

(i) 2 + 4 + 6 + 8 + ——- + 20 is a finite series.

(ii) 1 + 3 + 5 + ——– is an infinite series.

Progression :

It is not necessary that the terms of a sequence always follow a certain pattern or they are described by some explicity formula for the nth term. Those sequence whose terms follow certaion patterns are called progression.

OR

It the terms of a sequence are written under specific conditions , then the sequence is called progression.

Arithmetic Progression (AP) :

A sequence is said to be an arithmetic progression , if the difference of a term and the term preceding to it is always same.

The constant difference , generally denoted by d , is called the common difference.

An arithmetic progression (AP) is a sequence whose terms increase or decrease by a fixed number. This fixed number is called the common difference of the A.P. , generally denoted by d.

In other word ,Â

IfÂ  a1 , a2 , a3 , ——- , an are in A.P. then a2 – a1 = a3 – a2 = ——- = anÂ  – an -1 = d (say)

If a is the first term and d is the common difference , then A.P. can be written as

a , a + d , a + 2 d , ———- , { a + (n – 1) d } , ——- .

Example :Â

(i) 1 , 3 , 5 , 7 , ——–Â

(ii) 2 , 4 , 6 , ———

The nth Term of an A.P. :

Let a be the first term , d be the common difference and l be the last term of an A.P. then nth term is given byÂ

Tn = a + (n -1) dÂ  whereÂ  d = Tn – Tn -1Â

The nth term from last isÂ  T ‘n = l – (n -1) d

Example : How many terms are there in the A.P.Â  20 , 25 , 30 , —— , 100 ?

(1) 15Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â

(2) 17Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â

(3) 12Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â

(4) 14

Solution : (2) LetÂ  the number of terms be n.

Given , Tn = 100Â  , a = 20 , d = 5Â

We know ,

\begin{aligned} & T_n=a+(n-1) d \\ & 100=20+(n-1) 5 \\ & 100-20=(n-1) 5 \\ & n-1=\frac{80}{5} \\ & n-1=16 \\ & n=17 \end{aligned}

âˆ´Â  Â Number of terms = 17Â

The sum of n Terms of an A.P. :Â

Suppose there are n terms of a sequence , whose first term is a , common difference is d and last term is l , then sum of n terms is given byÂ

\begin{aligned} S_n & =\frac{n}{2}[2 \mathrm{a}+(\mathrm{n}-1) \mathrm{d}] \\ & =\frac{n}{2}[\mathrm{a}+\mathrm{l}] \end{aligned}

i.e. Sn is called as first n term of an A.P.

Example : The sum of all natural numbers lying between 100 and 1000 , which are multiples of 5 isÂ

(1) 98400Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â

(2) 98450Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â

(3) 98436Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â

(4) 98455

Solution : (2) The numbers are 105 , 110 , 115Â  ,Â  ——– , 995.

Here , first term (a) = 105 & common difference (d) = 110 – 105 = 5

Now , nth term ,

\begin{aligned} & T_n=a+(n-1) d \\ & 995=105+(n-1) 5 \\ & 995-105=(n-1) 5 \\ & n-1=\frac{890}{5} \\ & n-1=178 \\ & n=179 \end{aligned}

Now, the required sum,
\begin{aligned} S_n & =\frac{\mathrm{n}}{2}[a+l] \\ & =\frac{179}{2}[105+995] \\ & =\frac{179}{2} \times 1100 \\ & =179 \times 550 \\ & =98450 \end{aligned}

Important Results Related to A.P. :Â
• For any sequence : Tn = SnÂ  – Sn – 1Â
• For an A.P. : d = Sn – 2 Sn – 1 + Sn – 2
• IfÂ  ( a1 , a2 , a3 , ——- , an )Â  â†’ A.P. then ( a1 Â± kÂ  , a2 Â± k , a3 Â± k , ——- , an Â± k ) â†’ A.P.
• IfÂ  ( a1 , a2 , a3 , ——- , an )Â  â†’ A.P. then ( k a1 , k a2 , k a3 , ——- , k anÂ ) â†’ A.P.
• If a , b , c are in A.P. â‡’ 2 b = a + cÂ
• 3 terms in A.P. â‡’ a – d , a , a + d
• 4 terms in A.P. â‡’ a – 3 d , a – d , a + d , a + 3 dÂ
• 5 terms in A.P. â‡’ a – 2 d , a – d , a , a + d , a + 2 dÂ
• Sum of terms of an A.P. equidistance from starting and end is equal.

i.e. IfÂ  a1 , a2 , a3 , ——- a42 , a43Â  â†’ A.P. thenÂ  a1 + a43 = a2 + a42 = a3 + a41 = ——-

Example : If a1 , a2 , a3 , —– , an are in A.P. and a1 + a2 + a3 + ——- + an = 114 then a1 + a6 + a11 + a16 is equal to :

(a) 98Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â

(b) 38Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â

(c) 64Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â

(d) 76

Solution : (d)Â  a1 + a2 + a3 + ——- + an = 114 thenÂ

a1 + a16 = x ( Let )

a4 + a13 = x

a7 + a10 = xÂ  thenÂ

x + x + x = 114Â

3 x = 114Â

x = 38Â

âˆ´ a1 + a6 + a11 + a16

Â = (a1 + a16 ) + (a6 + a11 )Â

= x + xÂ

= 2 xÂ

= 2 Ã— 38

= 76

FAQs :Â
Answer : A sequence is defined as an arrangement of numbers in a particular order. On the other hand , a series is defined as the sum of the elements of a sequence.
Answer : A few popular sequences in maths are : Arithmetic sequences Geometric Sequences Harmonic Sequences Fibonacci Numbers
Answer : An example of sequence : 2 , 4 , 6, 8 , ---- An example of a series : 2 + 4 + 6 + 8 + ------
Answer : The formula to determine the common difference in an Arithmetic sequence is : d = Successive term - Preceding term
Answer : If "a" is the first term and "d" is the common difference of an arithmetic sequence , then it is represented by a , a + d < a + 2 d , a + 3 d , ----

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