Revision Notes on Sequence and Series

• A sequence is a function of natural numbers with codomain as the set of real numbers or complex numbers.

• A sequence may be finite or infinite depending on whether it has finite or infinite terms.

• A progression is a sequence having its terms in a definite pattern.

• A Series is constructed by adding or subtracting the terms of a sequence.   

• A series having finite number of terms is called a finite series while the series having infinite number of terms is called infinite sereis.

Arithmetic progression and Arithmetic Mean

• If ‘a’ is the first term and ‘d’ is the common difference of the arithmetic progression then its n^th term is given by a_n = a + (n-1)d

• In a_n = a + (n-1)d, note that:

if d > 0, then it is an increasing A.P.

if d < 0, then it is a decreasing A.P.

if d = 0, then all the terms remain same.

• The sum, S_n of the first ‘n’ terms of the A.P. is given by S_n = n/2 [2a + (n-1)d]

• If S_n is the sum of n terms of an A.P. whose first term is ‘a’ and last term is ‘l’,S_n = (n/2)(a + l).

• If common difference is d, number of terms n and the last term l, then S_n = (n/2)[2l-(n -1)d]

• If a fixed number is added or subtracted from each term of an A.P., then the resulting sequence is also an A.P. and it has the same common difference as that of the original A.P. 

• If each term of A.P is multiplied by some constant or divided by a non-zero fixed constant, the resulting sequence is an A.P. again.

• The sum of two terms of an A.P. equidistant from the beginning and end is constant and equal to the sum of first and last terms.  

• If a₁, a₂, a₃, …, a_n and b₁, b₂, b₃, …, b_n, are in A.P. then a₁+b₁, a₂+b₂, a₃+b₃, ……, a_n+b_n and a₁–b₁, a₂–b₂, a₃–b₃, ……, a_n–b_n will also be in A.P.

• Suppose a₁, a₂, a₃, ……,a_n are in A.P. then a_n, a_n–1, ……, a₃, a₂, a₁ will also be in A.P.

• If n^th term of a series is t_n = An + B, then the series is in A.P.

• If a₁, a₂, a₃, ……, a_n are in A.P., then a₁ + a_n = a₂ + a_n–1 = a₃ + a_n–2 = …… and so on.

• In order to assume three terms in A.P. whose sum is given, they should be assumed as a-d, a, a+d.

• Four terms of the A.P. whose sum is given should be assumed as a-3d, a-d, a+d, a+3d

• Five convenient numbers in A.P. a–2b, a–b, a, a+b, a+2b.

• In general, we take a-rd, a-(r-1)d, …., a-d, a, a+rd in case we have to take (2r + 1) terms in an A.P. 

• Likewise, any 2r terms of an A.P. should be assumed as: a – (2r-1)d, a – (2r – 3)d, …., a – d, a, a + d, ………….. , a + (2r – 3)d, a + (2r – 1)d.

• The arithmetic mean of two numbers ‘a’ and ‘b’ is (a+b)/2.

• The terms A₁, A₂, ….. , A_n are said to be arithmetic means between a and b if a, A₁, A₂, ….. , A_n, b is an A.P.

• Clearly, ‘a’ is the first term, ‘b’ is the (n+2)^th term and ‘d’ is the common difference. Then, we have b = a+(n+2-1)d = a+(n+1)d

Hence, this gives ‘d’ = (b-a)/(n+1)

Geometric Progression and Geometric Mean

• If ‘a’ is the first term and ‘r’ is the common ratio of the geometric progression, then its n^th term is given by a_n = ar^n-1

• The sum, S_n of the first ‘n’ terms of the G.P. is given by S_n = a (rⁿ – 1)/ (r-1), when r ≠1; = na , if r =1

• If -1 < x < 1, then lim xⁿ = 0, as n →∞. Hence, the sum of an infinite G.P. is 1 + x + x²+ ….. = 1/(1-x)

• If -1 < r< 1, then the sum of the infinite G.P. is a +ar+ ar²+ ….. = a/(1-r)

• If each term of the G.P is multiplied or divided by a non-zero fixed constant, the resulting sequence is again a G.P.

• If each term of the G.P is raised to the same power then the resulting sequence is again a G.P

• If a₁, a₂, a₃, …. and b₁, b₂, b₃, … are two geometric progressions, then a₁b₁, a₂b₂, a₃b₃, …… is also a geometric progression and a₁/b₁, a₂/b₂, ... ... ..., a_n/b_n will also be in G.P.

• Suppose a₁, a₂, a₃, ……,a_n are in G.P. then a_n, a_n–1, a_n–2, ……, a₃, a₂, a₁ will also be in G.P.

• Taking the inverse of a G.P. also results a G.P. Suppose a₁, a₂, a₃, ……,a_n are in G.P then 1/a₁, 1/a₂, 1/a₃ ……, 1/a_n will also be in G.P

• If we need to assume three numbers in G.P. then they should be assumed as   a/b, a, ab  (here common ratio is b)

• Four numbers in G.P. should be assumed as a/b³, a/b, ab, ab³ (here common ratio is b²)

• Five numbers in G.P. a/b², a/b, a, ab, ab²  (here common ratio is b) 

• If a₁, a₂, a₃,… ,a_n is a G.P (a_i > 0 ∀i), then log a₁, log a₂, log a₃, ……, log a_n is an A.P. In this case, the converse of the statement also holds good.

• If three terms are in G.P., then the middle term is called the geometric mean (G.M.) between the two. So if a, b, c are in G.P., then b = √ac is the geometric mean of a and c.

• Likewise, if a₁, a₂, ……,a_n are non-zero positive numbers, then their G.M.(G) is given by G = (a₁a₂a₃ …… a_n)^1/n.

• If G₁, G₂, ……G_n are n geometric means between and a and b then a, G₁, G₂, ……,G_n b will be a G.P.

• Here b = arⁿ⁺¹ ⇒ r = ⁿ⁺¹√b/a, Hence, G₁ = a. ⁿ⁺¹√b/a, G₂ = a(ⁿ⁺¹√b/a)²,…, G_n = a(ⁿ⁺¹√b/a)ⁿ.

• Relation between A.M. and G.M. is A.M. ≥ G.M.

Arithmetic Geometric Progression 

• A sequence of the form S = a + (a+d)r + (a+2d)r² + (a+3d)r³ + …. is called an arithmetic geometric progression.

• Here, each term is the product of corresponding terms in an arithmetic and geometric series.

• T_n = [a + (n-1) d]r^n-1 is the nth term of A.G.P.

• Sum of n terms of A.G.P. is given by S = a/(1-r) + dr(1-r^n-1/(1-r)²) – [a+(n-1)d]rⁿ/(1-r), where r is the common ratio, d is the common difference and a is the first term.

• S_∞ = a/(1-r) + dr/(1-r)².

Harmonic Progression and Harmonic Mean

• Let a, b and c form an H.P. Then 1/a, 1/b and 1/c form an A.P.

• If a, b and c are in H.P. then 2/b = 1/a + 1/c,  which can be simplified as b = 2ac/(a+c).

• If ‘a’ and ‘b’ are two non-zero numbers then the sequence a, H, b is a H.P.

• The n numbers H₁, H₂, ……,H_n are said to be harmonic means between a and b, if a, H₁, H₂ ……, H_n, b are in H.P. i.e. if 1/a, 1/H₁, 1/H₂, ..., 1/H_n, 1/b are in A.P. Let d be the common difference of the A.P., Then 1/b = 1/a + (n+1) d ⇒ d = a–b/(n+1)ab.

  Thus 1/H₁ = 1/a + a–b/(n+1)ab,

          1/H₂ = 1/a + 2(a–n)/(n+1)ab,

               ………..     ……….

          1/H_n = 1/a + n(a–b)/(n+1)ab.

• If x₁, x₂, … x_n are n non-zero numbers, then the harmonic mean ‘H’ of these numbers is given by 1/H = 1/n (1/x₁ + 1/ x₂ + ……. +1/ x_n)

                                    

• As the nth term of an A.P is given by a_n = a + (n-1)d, So the nth term of an H.P is given by 1/ [a + (n -1) d].

• If we have a set of weights w₁, w₂, …. , w_n associated with the set of values x₁, x₂, …. , x_n, then the weighted harmonic mean is defined as

• Questions on Harmonic Progression are generally solved by first converting them into those of Arithmetic Progression.

• If ‘a’ and ‘b’ are two positive real numbers then A.M x H.M = G.M²

• The relation between the three means is defined as A.M > G.M > H.M.

• If we need to find three numbers in a H.P. then they should be assumed as 1/a–d, 1/a, 1/a+d

• Four convenient numbers in H.P. are 1/a–3d, 1/a–d, 1/a+d, 1/a+3d

• Five convenient numbers in H.P. are 1/a–2d, 1/a–d, 1/a, 1/a+d, 1/a+2d

Miscellaneous Sequences

• Sequences containing Σn, Σn², Σn³

^?   Σn = n(n+1)/2

   Σn² = n(n+1)(2n+1)/6

   Σn³ = [n(n+1)/2]²

• Using method of difference:

    If T₁, T₂,T₃ are the terms of sequence then the terms T₂ – T₁, T₃ – T₂, T₄ – T₃ …..may at times be in A.P. or sometimes may be in G.P. In such series, we first compute their nth term and then compute the sum to n terms, using sigma notation. 

• A series in which each term is composed of the reciprocal of the product of r factors in A.P., the first factor of the several terms being in the same A.P., then such series are solved by splitting the nth term as a difference of two.