CHAIN RULE

Progression of sequence and series
 (scroll down for class work solved questions)

• Arithmetic Progression (A.P.)  (or Arithmetic Sequence) 



Arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant, d to the preceding term. The constant d is called common difference.
An arithmetic progression is given by a, (a + d), (a + 2d), (a + 3d), ...
where a = the first term , d = the common difference
Examples for Arithmetic Progressions
1, 3, 5, 7, ... is an arithmetic progression (AP) with a = 1 and d = 2
7, 13, 19, 25, ... is an arithmetic progression (AP) with a = 7 and d= 6


• n^th term of an arithmetic progression
t_n = a + (n – 1)d

where t_n = n^th term, a= the first term , d= common difference
Example 1 : Find 10^th term in the series 1, 3, 5, 7, ...
a = 1
d = 3 – 1 = 2
10^th term, t₁₀ = a + (n-1)d = 1 + (10 – 1)2 = 1 + 18 = 19
Example 2 : Find 16^th term in the series 7, 13, 19, 25, ...
a = 7
d = 13 – 7 = 6
16^th term, t₁₆ = a + (n-1)d = 7 + (16 – 1)6 = 7 + 90 = 97



• Number of terms of an arithmetic progression
n=(l−a)d+1

where n = number of terms,

a= the first term ,

l = last term,

d= common difference
Example : Find the number of terms in the series 8, 12, 16, . . .72
a = 8
l = 72
d = 12 – 8 = 4
n=(l−a)d+1=(72−8)4+1=644+1=16+1=17



• Sum of first n terms in an arithmetic progression
Sn=n2[ 2a+(n−1)d ] =n2[ a+l ]where

a = the first term,

d= common difference, 
l=tn=nth term = a+(n−1)d
Example 1 : Find 4 + 7 + 10 + 13 + 16 + . . . up to 20 terms
a = 4
d = 7 – 4 = 3
Sum of first 20 terms, S₂₀ =n2[ 2a+(n−1)d ]=202[ (2×4)+(20−1)3 ]
=10[8+(19×3)]=10[8+57]=650



Example 2 : Find 6 + 9 + 12 + . . . + 30
a = 6
l = 30
d = 9 – 6 = 3
n=(l−a)d+1=(30−6)3+1=243+1=8+1=9
Sum, S = n2[ a+l ]=92[ 6+30 ]=92×36=9×18=162


• Arithmetic Mean
If a, b, c are in AP, b is the Arithmetic Mean (A.M.) between a and c.

In this case, b=12(a+c)
 
The Arithmetic Mean (A.M.) between two numbers a and b = 12(a+b)
 
If a, a₁, a₂ ... a_n, b are in AP we can say that a₁, a₂ ... a_n

are the n Arithmetic Means between a and b.
• If a, b, c are in AP, 2b = a + c
• To solve most of the problems related to A.P., the terms can be
  conveiently taken as
  
  3 terms : (a – d), a, (a +d)
  
  4 terms : (a – 3d), (a – d), (a + d), (a +3d)
  
  5 terms : (a – 2d), (a – d), a, (a + d), (a +2d)
• T_n = S_n - S_n-1
• If each term of an A.P. is increased, decreased , multiplied or
  
  divided by the same non-zero constant, the resulting sequence also
  
  will be in A.P.
• In an A.P., sum of terms equidistant from beginning and end will be constant



• Harmonic Progression (H.P.)  (or Harmonic Sequence)   



Non-zero numbers a1, a2, a3, ... an are in H.P. if 1a1, 1a2, 1a3,  ... 1an are in A.P.
Examples for Harmonic Progressions
12,16,110,⋯ is a harmonic progression (H.P.)
Three non-zero numbers a, b, c will be in HP, if  1a, 1b, 1c are in A.P.


If a, (a+d), (a+2d), . . . are in A.P., n^thterm of the A.P. = a + (n - 1)d

Hence, if 1a,1a+d,1a+2d,⋯ are in H.P., n^thterm of the H.P. = 1a+(n−1)d


If a, b, c are in HP, b is the Harmonic Mean(H.M.) between a and c

In this case, b=2aca+c


The Harmonic Mean (H.M.) between two numbers a and b = 2aba+b


If a, a₁, a₂ ... a_n, b are in H.P. we can say that a₁, a₂ ... a_n are the n Harmonic Means between a and b.
If a, b, c are in HP, 2b=1a+1c



• Geometric Progression (G.P.)  (or Geometric Sequence)   



A sequence of non-zero numbers is a Geometric Progression (G.P.) if the ratio of any term and its preceding term is always constant.
A Geometric Progression (G.P.) is given by a, ar, ar², ar³, ...
where a = the first term , r = the common ratio
Examples for Geometric Progressions
1, 3, 9, 27, ... is a geometric progression (G.P.) with a = 1 and r = 3
2, 4, 8, 16, ... is a geometric progression (G.P.) with a = 2 and r = 2


• n^th term of a geometric progression (G.P.)
tn=arn−1

where t_n = n^th term, a= the first term , r = common ratio, n = number of terms
Example 1 : Find the 10^th term in the series 2, 4, 8, 16, ...
a = 2,    r = 42 = 2,   n = 10
10^th term, t₁₀ = arn−1=2×210−1=2×29=2×512=1024


Example 2 : Find 5^th term in the series 5, 15, 45, ...
a = 5,   r = 155 = 3,   n = 5
5^th term, t₅ = arn−1=5×35−1 = 5 × 3⁴ = 5 × 81 = 405



• Sum of first n terms in a geometric progression (G.P.)
Sn=⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪a(rn−1)r−1 a(1−rn)1−r (if r > 1)(if r < 1)

where a= the first term , r = common ratio, n = number of terms



Example 1 : Find 4 + 12 + 36 + ... up to 6 terms
a = 4,   r = 124 = 3,   n = 6
Here r > 1. Hence, S6=a(rn−1)r−1=4(36−1)3−1=4(729−1)2=4×7282=2×728=1456


Example 2 : Find 1+12+14+ ... up to 5 terms
a = 1,   r = (12)1=12,   n = 5
Here r < 1. Hence, S6=a(1−rn)1−r=1[1−(12)5](1−12)=(1−132)(12)=(3132)(12)=3116=11516


• Sum of an infinite geometric progression (G.P.)
S∞=a1−r  (if    0 < r < 1)

where a= the first term , r = common ratio


Example : Find 1+12+14+18+ . . . ∞
a = 1,       r = (12)1=12
Here 0 < r < 1. Hence, S∞=a1−r=1(1−12)=1(12)=2
• Geometric Mean
If three non-zero numbers a, b, c are in G.P., b is the Geometric Mean (G.M.) between a and c.

In this case, b=ac−−√
The Geometric Mean (G.M.) between two numbers a and b = ab−−√

(Note that if a and b are of opposite sign, their G.M. is not defined.)
• If a, b, c are in G.P., b² = ac
• If a, b, c are in G.P., a−bb−c=ab
• In a G.P., product of terms equidistant from beginning and end will be constant.
• To solve most of the problems related to G.P., the terms of the G.P. can be conveiently taken as
  
  3 terms : ar, a, ar
  
  5 terms : ar2, ar, a, ar, ar²



• Relationship Between Arithmetic Mean, Harmonic Mean,
   and Geometric Mean of Two Numbers
  
     



If GM, AM and HM are the Geometic Mean, Arithmetic Mean and Harmonic Mean of two positive numbers respectively, then

GM² = AM × HM

• Some Interesting Properties to Note   



Three numbers a, b and c are in AP if b=a+c2

Three non-zero numbers a, b and c are in HP if b=2aca+c

Three non-zero numbers a, b and c are in HP if a−bb−c=ac


Let A, G and H be the A.M., G.M. and H.M. between two distinct positive numbers. Then


• A > G > H


• A, G and H are in GP


If a series is both an A.P. and G.P., all terms of the series will be equal. In other words, it will be a constant sequence.



• Power Series : Important formulas    

• 1+1+1+⋯ n terms=∑1=n
• 1+2+3+⋯+n=∑n=n(n+1)2
• 12+22+32+⋯+n2=∑n2=n(n+1)(2n+1)6
• 13+23+33+⋯+n3=∑n3=n2(n+1)24=[n(n+1)2]2