Sequences and series
(A) Arithmetic Progression
A list of numbers in a specific order is known as a sequence or progression.
Let
f(n) = 4n + 1
For n = 1, f(n) = 5
For n = 2, f(n) = 9
For n = 3, f(n) = 13
For n = 4, f(n) = 17
So sequence or progression will be
5, 9, 13, 17......
An Arithmetic Progression or Arithmetic sequence (AP) is a sequence of numbers such that the difference(Common Difference of AP) between any two consecutive terms is constant.
Given progression is in AP
A1, A2, A3, A4, A5, A6, A7, A8
If and only if
A2-A1= A3-A2 = A4-A3 = A5-A4 = A6-A5 = A7-A6 = A8 - A7
Note : Each number in the sequence is known as term.
Example 1
2, 4, 6, 8, 10, 12, 14, ....................40
Here
4-2 = 6-4 = 8- 6 = 10-8 = 2
(B) Formula For Common Difference(d) of AP
Let any AP
A1, A2, A3, A4, A5, ........An-1, An, An+1 ......
d = A2 - A1 = A3 - A4 = An - An-1
So Common difference d of an AP is calculated by
d = (n)th term - (n -1)th term
Or
d = (n+1)th term - (n)th term
Example 2
Let first term of an AP = a
Let Common Difference ( Difference between two consecutive terms of an AP) = d
So,
First term = a = a + (1-1)d
Second term = a + d = a + (2-1)d
Third term = a + d + d = a + (3-1)d
Fourth term = a +2d + d = a + (4-1)d
Fifth term = a + 3d + d = a + (5-1)d
....... ......
........ .........
........ ........
nTh term , Tn = a + (n - 1)d
So Arithmetic Progression can be written as
Or General form of an AP written as
a, a +d, a + 2d, a + 3d, a + 4d, a + 5d, .........
.......... a + (n - 1)d , a + n d , a +(n +1)d ... .. infinity
Here,
a = Fist term of AP
d = Common Difference between two consecutive terms of AP
n = Number of terms of AP
Tn = nth term of AP = a + (n - 1)d
(C) nth Term of Arithmetic Progression
Tn = a + (n - 1)d
Here,
a = Fist term of AP
d = Common Difference between two consecutive terms of AP
n = Number of terms of AP
If the initial term of an arithmetic progression is "a" and the common difference of successive members is "d' then the nth term (Tn) of the sequence is given by:
Tn = a + (n - 1)d
If there are m terms in the AP, if first term is "a" and common difference of two consecutive term is "d" then last term( Tm) is given by:
Tm = a + (m - 1)d
Note :
(1) If d is Positive, AP terms will grow towards infinity
(2) If d is Negative, AP terms will grow towards negative infinity
(D) Types of Arithmetic Progression
There are two type of Arithmetic Progression
(1) Finite Arithmetic Progression: An Arithmetic Progression containing a finite number of terms is called finite Arithmetic Progression. A finite AP has a last term.
For example: 4, 8, 12, 16, 20, .... 36, 40
(2) Infinite Arithmetic Progression : An Arithmetic Progression which does not have a finite number of terms is called infinite Arithmetic Progression. Such APs do not have a last term.
1, 2, 3, 4, 5, 6, 7, 8, 9, ....................
(E) mth term of an AP from the end
Let an AP have total numbers of term is "n"
So
mth term of AP from end = (n-m +1)th term of AP from the beginning
Let me verify the above the above concept
Let an AP has total number of term = 10 like as below
A1, A2, A3, A4, A5, A6, A7, A8, A9, A10
So,
2nd term from end = A9 = 9th term from beginning = (10-2+1)th term from beginning
3rd term from end = A8 = 8th term from beginning = (10-3+1)th term from beginning
6nd term from end = A5 = 5th term from beginning = (10-6+1)th term from beginning
8th term from end = A3 = 3rd term from beginning = (10-8+1)th term from beginning
(F) Arithmetic Series
The arithmetic series is the sum of all the terms of the arithmetic sequence.
(A) Arithmetic Progression
A list of numbers in a specific order is known as a sequence or progression.
Let
f(n) = 4n + 1
For n = 1, f(n) = 5
For n = 2, f(n) = 9
For n = 3, f(n) = 13
For n = 4, f(n) = 17
So sequence or progression will be
5, 9, 13, 17......
An Arithmetic Progression or Arithmetic sequence (AP) is a sequence of numbers such that the difference(Common Difference of AP) between any two consecutive terms is constant.
Given progression is in AP
A1, A2, A3, A4, A5, A6, A7, A8
If and only if
A2-A1= A3-A2 = A4-A3 = A5-A4 = A6-A5 = A7-A6 = A8 - A7
Note : Each number in the sequence is known as term.
Example 1
2, 4, 6, 8, 10, 12, 14, ....................40
Here
4-2 = 6-4 = 8- 6 = 10-8 = 2
(B) Formula For Common Difference(d) of AP
Let any AP
A1, A2, A3, A4, A5, ........An-1, An, An+1 ......
d = A2 - A1 = A3 - A4 = An - An-1
So Common difference d of an AP is calculated by
d = (n)th term - (n -1)th term
Or
d = (n+1)th term - (n)th term
Example 2
Let first term of an AP = a
Let Common Difference ( Difference between two consecutive terms of an AP) = d
So,
First term = a = a + (1-1)d
Second term = a + d = a + (2-1)d
Third term = a + d + d = a + (3-1)d
Fourth term = a +2d + d = a + (4-1)d
Fifth term = a + 3d + d = a + (5-1)d
....... ......
........ .........
........ ........
nTh term , Tn = a + (n - 1)d
So Arithmetic Progression can be written as
Or General form of an AP written as
a, a +d, a + 2d, a + 3d, a + 4d, a + 5d, .........
.......... a + (n - 1)d , a + n d , a +(n +1)d ... .. infinity
Here,
a = Fist term of AP
d = Common Difference between two consecutive terms of AP
n = Number of terms of AP
Tn = nth term of AP = a + (n - 1)d
(C) nth Term of Arithmetic Progression
Tn = a + (n - 1)d
Here,
a = Fist term of AP
d = Common Difference between two consecutive terms of AP
n = Number of terms of AP
If the initial term of an arithmetic progression is "a" and the common difference of successive members is "d' then the nth term (Tn) of the sequence is given by:
Tn = a + (n - 1)d
If there are m terms in the AP, if first term is "a" and common difference of two consecutive term is "d" then last term( Tm) is given by:
Tm = a + (m - 1)d
Note :
(1) If d is Positive, AP terms will grow towards infinity
(2) If d is Negative, AP terms will grow towards negative infinity
(D) Types of Arithmetic Progression
There are two type of Arithmetic Progression
(1) Finite Arithmetic Progression: An Arithmetic Progression containing a finite number of terms is called finite Arithmetic Progression. A finite AP has a last term.
For example: 4, 8, 12, 16, 20, .... 36, 40
(2) Infinite Arithmetic Progression : An Arithmetic Progression which does not have a finite number of terms is called infinite Arithmetic Progression. Such APs do not have a last term.
1, 2, 3, 4, 5, 6, 7, 8, 9, ....................
(E) mth term of an AP from the end
Let an AP have total numbers of term is "n"
So
mth term of AP from end = (n-m +1)th term of AP from the beginning
Let me verify the above the above concept
Let an AP has total number of term = 10 like as below
A1, A2, A3, A4, A5, A6, A7, A8, A9, A10
So,
2nd term from end = A9 = 9th term from beginning = (10-2+1)th term from beginning
3rd term from end = A8 = 8th term from beginning = (10-3+1)th term from beginning
6nd term from end = A5 = 5th term from beginning = (10-6+1)th term from beginning
8th term from end = A3 = 3rd term from beginning = (10-8+1)th term from beginning
(F) Arithmetic Series
The arithmetic series is the sum of all the terms of the arithmetic sequence.
The arithmetic series is in the form of
{a + (a + d) + (a + 2d) + (a + 3d) + .................}
(G) Sum of n terms of an Arithmetic Series
Let an AP has n terms
first term = a
Last term = l = a + (n -1)d
Sum, Sn = n/2( a + l)
Sn = n/2( 2a+(n-1)d)
(H) Important Note
1. The sum of the infinite arithmetic sequence does not exist.
2. Sum of first n natural numbers
= n(n+1)/2
3. Sum of first n odd natural number = n^2
4. Sum of first n even natural numbers = n(n+1)
5. Sum of Square of first n natural numbers = n(n+1)(2n+1)/6
6. Sum of Cubes of first n natural numbers = [ n(n+1)/2]^2
7. sum of the fourth power of n natural numbers
= n(n +1)(2n +1)( 3n^2 +3n -1)/30
8. If rth term of an AP is
Tr = Ar^3 + Br^2 + Cr + D
then
Sum of n terms of that AP will be
Sn = Summation r =1 to r =n (Tr)
9. If in an AP pth term is q and qth term is p , then rth term will be = p + q - r
10. If in an AP sum of p terms is q and sum of q terms is p so, sum of (p+q) terms will be = -(p+q)
11. If in an AP sum of p terms is equal to sum of q terms then sum of (p +q) term will be Zero (0)
12. For an Arithmetic Series if sum of n terms given as Sn so we can calculate nth term Tn as Tn = Sn - Sn-1
13. Common Difference(d) of an AP can be calculated by d = S2 - 2S1
where S2 = Sum of first 2 terms of that AP
S1 = Sum of First term of that AP
14. Sum of infinite Arithmetic progression is infinity if d is > 0 and Sum of Infinite Arithmetic progression - Infinity if d is <0
15. If sum of n terms of an AP is in the form
Sn = An^2 + Bn +C ( Quadratic form )
So Common Difference(d) will be = 2 × Coefficient of n ^2
d = 2A
16. If nth term Tn of an AP is in the form of
Tn = Pn + Q ( Linear Expression )
Common difference(d) = Coefficient of n = P
17. Let there are two AP, AP1 and AP2
Let sum of n terms of AP1 = S1n = f1(n)
Let sum of n terms of AP2 = S2n = f2(n)
And if
S1n/S2n = f1(n)/f2(n)
Then
T1(n)/T2(n) = f1(2n-1)/f2(2n-1)
Here
T1n = nth term of AP1
T2n = nth term of AP2
18. Let if T1n/T2n = f1(n)/f2(n)
Then S1n / S2n = f1[(n+1)/2]/f2[(n+1)/2]
(I) Arithmetic Mean
(1) Arithmetic mean between two numbers p and q will be AM = (p +q)/2
p , AM, q .... are in AP
AM - p = q - AM
2AM = p + q
AM = (p +q)/2
(2) Arithmetic mean of n numbers a1, a2, a3, .....an
AM = (a1 + a2 + a3 + .... an)/n
(J) n Arithmetic Mean between two numbers p and q
(1)
p, A1, A2, A3, ........ An, q are in AP
First term(a)= p
Total number of term(N) = n +2
Last term(Tn) = q
So,
Tn = a + (N-1)d
d = Common Difference
q = p + (n +2 -1)d
q - p = (n+1)d
d = (q-p)/(n+1)
A1 = p + d = p + (q-p)/(n+1)
A2 = p + 2d = p + 2(q-p)/(n+1)
.........
...
..
An = p + nd = p + n(q-p)/(n+1) = q - d
(2) Sum of n AM between p and q
A1 + A2 + A3 .....+ An = n(p+q)/2
(3) Sum of r AM between p and q
A1 + A2 + A3 ...... + Ar = r(p+q)/2
(4) Sum of n AM between p and q / Sum of r AM between p and q = [n(p+q)/2]/[r(p+q)/2] = n/r
(5) Select 3 terms in AP
a - d , a , a + d
(6) Select 5 terms in AP
a - 2d, a - d, a, a +d, a + 2d
(7) Select 7 terms in AP
a - 3d, a - 2d, a - d, a, a +d, a + 2d, a + 3d
(8) Select 4 terms in AP
a - 3d, a - d, a + d, a + 3d
(9) Select 6 terms in AP
a - 5d, a - 3d, a - d, a + d, a + 3d, a + 5d
(10) Select 8 terms in AP
a - 7d, a - 5d, a - 3d, a - d, a + d, a + 3d, a + 5d, a + 7d
(11) For a series, if mth term in the form of T(m) = Cm + D, Then this Series is in Arithmetic Progression
(12) For a series if sum of m terms in the form of S(m) = Cm^2 + Dm + E, then this Series will be in AP
(13) T(m) = Cm + D, = mth term of a Series ( AP )
Then T(m/k), T(mk), T(m+k), T(m-k) ..Will be in AP k is a non zero constant
{a + (a + d) + (a + 2d) + (a + 3d) + .................}
(G) Sum of n terms of an Arithmetic Series
Let an AP has n terms
first term = a
Last term = l = a + (n -1)d
Sum, Sn = n/2( a + l)
Sn = n/2( 2a+(n-1)d)
(H) Important Note
1. The sum of the infinite arithmetic sequence does not exist.
2. Sum of first n natural numbers
= n(n+1)/2
3. Sum of first n odd natural number = n^2
4. Sum of first n even natural numbers = n(n+1)
5. Sum of Square of first n natural numbers = n(n+1)(2n+1)/6
6. Sum of Cubes of first n natural numbers = [ n(n+1)/2]^2
7. sum of the fourth power of n natural numbers
= n(n +1)(2n +1)( 3n^2 +3n -1)/30
8. If rth term of an AP is
Tr = Ar^3 + Br^2 + Cr + D
then
Sum of n terms of that AP will be
Sn = Summation r =1 to r =n (Tr)
9. If in an AP pth term is q and qth term is p , then rth term will be = p + q - r
10. If in an AP sum of p terms is q and sum of q terms is p so, sum of (p+q) terms will be = -(p+q)
11. If in an AP sum of p terms is equal to sum of q terms then sum of (p +q) term will be Zero (0)
12. For an Arithmetic Series if sum of n terms given as Sn so we can calculate nth term Tn as Tn = Sn - Sn-1
13. Common Difference(d) of an AP can be calculated by d = S2 - 2S1
where S2 = Sum of first 2 terms of that AP
S1 = Sum of First term of that AP
14. Sum of infinite Arithmetic progression is infinity if d is > 0 and Sum of Infinite Arithmetic progression - Infinity if d is <0
15. If sum of n terms of an AP is in the form
Sn = An^2 + Bn +C ( Quadratic form )
So Common Difference(d) will be = 2 × Coefficient of n ^2
d = 2A
16. If nth term Tn of an AP is in the form of
Tn = Pn + Q ( Linear Expression )
Common difference(d) = Coefficient of n = P
17. Let there are two AP, AP1 and AP2
Let sum of n terms of AP1 = S1n = f1(n)
Let sum of n terms of AP2 = S2n = f2(n)
And if
S1n/S2n = f1(n)/f2(n)
Then
T1(n)/T2(n) = f1(2n-1)/f2(2n-1)
Here
T1n = nth term of AP1
T2n = nth term of AP2
18. Let if T1n/T2n = f1(n)/f2(n)
Then S1n / S2n = f1[(n+1)/2]/f2[(n+1)/2]
(I) Arithmetic Mean
(1) Arithmetic mean between two numbers p and q will be AM = (p +q)/2
p , AM, q .... are in AP
AM - p = q - AM
2AM = p + q
AM = (p +q)/2
(2) Arithmetic mean of n numbers a1, a2, a3, .....an
AM = (a1 + a2 + a3 + .... an)/n
(J) n Arithmetic Mean between two numbers p and q
(1)
p, A1, A2, A3, ........ An, q are in AP
First term(a)= p
Total number of term(N) = n +2
Last term(Tn) = q
So,
Tn = a + (N-1)d
d = Common Difference
q = p + (n +2 -1)d
q - p = (n+1)d
d = (q-p)/(n+1)
A1 = p + d = p + (q-p)/(n+1)
A2 = p + 2d = p + 2(q-p)/(n+1)
.........
...
..
An = p + nd = p + n(q-p)/(n+1) = q - d
(2) Sum of n AM between p and q
A1 + A2 + A3 .....+ An = n(p+q)/2
(3) Sum of r AM between p and q
A1 + A2 + A3 ...... + Ar = r(p+q)/2
(4) Sum of n AM between p and q / Sum of r AM between p and q = [n(p+q)/2]/[r(p+q)/2] = n/r
(5) Select 3 terms in AP
a - d , a , a + d
(6) Select 5 terms in AP
a - 2d, a - d, a, a +d, a + 2d
(7) Select 7 terms in AP
a - 3d, a - 2d, a - d, a, a +d, a + 2d, a + 3d
(8) Select 4 terms in AP
a - 3d, a - d, a + d, a + 3d
(9) Select 6 terms in AP
a - 5d, a - 3d, a - d, a + d, a + 3d, a + 5d
(10) Select 8 terms in AP
a - 7d, a - 5d, a - 3d, a - d, a + d, a + 3d, a + 5d, a + 7d
(11) For a series, if mth term in the form of T(m) = Cm + D, Then this Series is in Arithmetic Progression
(12) For a series if sum of m terms in the form of S(m) = Cm^2 + Dm + E, then this Series will be in AP
(13) T(m) = Cm + D, = mth term of a Series ( AP )
Then T(m/k), T(mk), T(m+k), T(m-k) ..Will be in AP k is a non zero constant
👍1
(14) If each term of an AP will multiplied, divided, Increased or Decreased by a non zero constant k .. then resulting Sequence will also be in AP
(15 ) For a Series if sum of m terms is S(m) = Cm^2 + Dm + E,
Then S(m/k), S(mk), S(m + k ), S(m - k) will be in AP, k is a non zero constant
(16) If A1, A2, A3, A4, Are in AP
Then 1/A1, 1/A2, 1/A3, 1/A4 are in HP (Harmonic Progression)
(17) The sum of any two terms equidistant from the beginning and end of an Arithmetic progression is always constant. Furthermore, their sum is always equal to the sum of the first and last terms.
(15 ) For a Series if sum of m terms is S(m) = Cm^2 + Dm + E,
Then S(m/k), S(mk), S(m + k ), S(m - k) will be in AP, k is a non zero constant
(16) If A1, A2, A3, A4, Are in AP
Then 1/A1, 1/A2, 1/A3, 1/A4 are in HP (Harmonic Progression)
(17) The sum of any two terms equidistant from the beginning and end of an Arithmetic progression is always constant. Furthermore, their sum is always equal to the sum of the first and last terms.
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National Testing Agency (NTA) will soon release the application date for JEE Main 2023. The registration process for the first session of JEE Main January will start from the last week of November. After November 20, NTA will release the notification. At the same time, the application process for JEE Main April session will be started in February 2023. Students enrolling in engineering can participate in the JEE Main registration process.
NTA has asked to keep an eye on jeemain.nta.nic.in. In JEE Main 2023 also the question paper will be divided into two parts. Part A will consist of multiple choice questions, while Part B will consist of questions of numerical value. Part A is compulsory and four marks will be awarded for every correct answer, while one mark will be deducted for every wrong answer. Whereas in Part B, out of 10 questions, any five questions will have to be answered. There will be no negative marking in Part B.
Tomorrow on 7th Of November You was born and We remember you for your Groundbreaking Discovery which known as "Raman Effect" (28 Feb 1928) later on we were celebrate this day as "National Science Day"| After Your Discovery You was nominated for Nobel Prize and you Honoured with this Prize in 1930 ❤
Regards
Er. Gyanesh Rai
Mathematics Expert
Radius Jee
L-3/87 Sector - D Puraniya Chauraha Aliganj Lucknow
05224312655
07042982879
Regards
Er. Gyanesh Rai
Mathematics Expert
Radius Jee
L-3/87 Sector - D Puraniya Chauraha Aliganj Lucknow
05224312655
07042982879
Q1. Calculate the value of I
If I = [ log₂35 ]
Where [ x ] Denotes greatest integer function less than or equal to x
If I = [ log₂35 ]
Where [ x ] Denotes greatest integer function less than or equal to x
Anonymous Quiz
35%
(A) 5
26%
(B) 6
26%
(C) 7
12%
(D) Non of the above
Q2. Solve for x ?
if x[x] = 15 [x] Denotes the greatest integer less than or equal to x
if x[x] = 15 [x] Denotes the greatest integer less than or equal to x
Anonymous Quiz
36%
(A) 5
30%
(B) 3
12%
(C) 4
22%
(D) None of the above
If
Lim x→0 f(x) EXIST
That Means
Left Hand Limit (LHL) = Right Hand Limit(RHL) at x =0
LHL Means when x approaches from left hand side of zero on real number line or on x axis
In this case we assume that x is slightly less than ZERO(0)
LHL Represented by
LHL = x→0⁻ f(x)
RHL Means when x approaches from Right hand side of zero on real number line or on x axis
In this case we assume that x is slightly greater than ZERO(0)
RHL Represented by
RHL = x→0⁺ f(x)
NOTE 1
If
x→0⁻ f(x) = = x→0⁺ f(x)
in this case LIMIT WILL EXIST AT X =0
NOTE 2
If
x→0⁻ f(x) = = x→0⁺ f(x) = f(0)
f(0) = value of function at x =0
in this case FUNCTION f(x) WILL BE CONTINUOUS AT X =0
THANKS
Best Wishes
Lim x→0 f(x) EXIST
That Means
Left Hand Limit (LHL) = Right Hand Limit(RHL) at x =0
LHL Means when x approaches from left hand side of zero on real number line or on x axis
In this case we assume that x is slightly less than ZERO(0)
LHL Represented by
LHL = x→0⁻ f(x)
RHL Means when x approaches from Right hand side of zero on real number line or on x axis
In this case we assume that x is slightly greater than ZERO(0)
RHL Represented by
RHL = x→0⁺ f(x)
NOTE 1
If
x→0⁻ f(x) = = x→0⁺ f(x)
in this case LIMIT WILL EXIST AT X =0
NOTE 2
If
x→0⁻ f(x) = = x→0⁺ f(x) = f(0)
f(0) = value of function at x =0
in this case FUNCTION f(x) WILL BE CONTINUOUS AT X =0
THANKS
Best Wishes
Q3. Calculate the value of I
if I = [√1] + [√2] + [√3] .......[√100]
Where [ x ] Denotes greatest integer less than or equal to x.
if I = [√1] + [√2] + [√3] .......[√100]
Where [ x ] Denotes greatest integer less than or equal to x.
Anonymous Quiz
20%
(A) 615
30%
(B) 625
42%
(C) 5050
8%
(D) 505
Q4. Solve for x
If f₁(x) = f₂(x), Given
f₁(x) = [x/2] + [(x+1)/2] f₂(x) = [x/2022] + [(x+1)/2022] + [(x+2)/ 2022] + ............+ [(x+2021)/2022] Where [ ● ] Denotes Greatest Integer function less than or equal to x,
If f₁(x) = f₂(x), Given
f₁(x) = [x/2] + [(x+1)/2] f₂(x) = [x/2022] + [(x+1)/2022] + [(x+2)/ 2022] + ............+ [(x+2021)/2022] Where [ ● ] Denotes Greatest Integer function less than or equal to x,
Anonymous Quiz
12%
(A) x ∈ [0 ∞)
41%
(B) x ∈ (0 ∞)
42%
(C) x ∈ (-∞ ∞)
5%
(D) None of These
Q5.If D₁ is the domain of the function f₁(x) = √x² - [x] ² ) and D₂ is the domain of function f₂(x) = 1/f₁(x) calculate the value of D₁ ∩ D₂ ? Where Denotes greatest integer function less than or equal to x and Z denotes Integers【RADIUS JEE】
Anonymous Quiz
29%
(A) x ∈ (-∞ ∞)
46%
(B) x ∈ (-∞ ∞) - Z
19%
(C) x ∈ (0 ∞) - Z
6%
(D) None of the above