Short Notes on Series and Progressions

Solution:

Such sums are done by substituting n = 1, 2, 3, … one by one. 

Here t = n² (n + 1)

Put n = 1, 2, 3. So the series is 2, 12, 36, … 

REPRESENTATION OF SEQUENCE 

Sequences can be represented in various ways.

For example, 1, 3, 5, … is a sequence which can be represented by the formula: (2n – 1).

It can also be represented by a rule of writing the nth term of the sequence. For example, the sequence 1, 3, 5, 7, … can be written as a_n = 2n – 1.

Sometimes we represent a real sequence by using a recursive relation.

For example, the Fibonacci sequence is given by a₁ = 1, a₂ = 1 and a_{n + 1} = a_n + a_{n – 1}, n ≥ 2. The terms of this sequence are 1, 1, 2, 3, 5, 8, … 

SERIES

If a₁, a₂, a₃, a₄, …, a_n, … is a sequence, then the expression a₁ + a₂ + a₃ + a₄ + a₅ + … + a_n + … is a series.

A series is finite or infinite according to as the number of terms in the corresponding sequence is finite or infinite.

PROGRESSIONS

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

ARITHMETIC PROGRESSION (AP)

A sequence of numbers is said to be in Arithmetic progression if each number differs from its succeeding number by a constant value. 

Common difference: Take the series: 1, 4, 7, 10, ……In this, every term increases by the same number every time. Any term deducted from the succeeding term results in the same difference of 3. (4 - 1 = 3 and 7 - 4 = 3). This is called the common difference and is denoted by 'd'. 

The AP could also have a negative common difference: 6, 3, 0, -3…..each term is reduced by 3 successively. Here the common difference is -3. 

The AP can be denoted by a, a + d, a + 2d, a + 3d ……where ‘a’ is the first term and ‘d’ is the common difference. 

If the number of terms in the sequence = n and its first term is a, then nth term of the AP is given by: a + (n - 1) d where 'd' is the common difference. 

The sum of n terms of an AP is given by the equation

Illustration 2: Find the 20th term of series 4, 7, 10…… and also the sum of the 20 terms.

Solution:Here a = 4, d = 3, n = 20. 

20^th term is given by a + (n – 1)d ⇒ 4 + 19 × 3 = 61. Sum of 20 terms = 20/2 × [8 + 19 × 3] = 650. 

Illustration 3: The sum of a certain number of terms of an A.P. is 57 and the first and last terms are 17 and 2 respectively. Find the number of terms and the common difference of the series.

Solution: Here S = 57, a = 17, l = 2.

Using the formula S = n/2 (a + l) we get 57 = n/2 (17 + 2) ⇒ 114 = 19n, or n = 6. 

Since 6^th terms is 2, we can use the formula for nth term: a + (n – 1)d ⇒ 2 = 17 + 5 × d; On solving we get 5d = - 15 and d = - 3. 

Illustration 4: The ratio of the sum of n terms of two A.P.’s is (7n + 1) : (4n + 27). Find the ratio of their mth terms. 

Solution: Let a₁, a₂ be the first terms and d₁, d₂ the common differences of the two given A.P.’s. Then sums of their n terms are given by

Illustration 6: The ratio of the sums of m and n terms of an A.P. is m2 : n2. Find the ratio of the mth and nth terms

Solution:

GEOMETRIC PROGRESSION (GP)

A sequence of numbers is said to be in Geometric Progression if there exists a constant factor such that any number in the sequence when multiplied by this constant factor results in the next number of the sequence. 

For example, in the series 1, 3, 9, 27, …… , each term is multiplied by 3 to give the succeeding term. This is called the common ratio: each term divided by the preceding term will give us 3. In the series, 4, 2, 1, ½ …… any term multiplied by ½ gives the succeeding term (4 × ½ = 2) so the common ratio is 1/2. 

If the first term is 'a' and the common ratio is 'r', then the series is a, ar, ar², ar³, ……..ar^{n - 1}. 

The nth term of a GP is given by a. r ^{(n- 1)}

The sum of terms upto infinity of a GP for which r < 1, is given by: a/(1-r)

Some important results:
1. Sum of 'n' natural numbers: 1 + 2 + 3 + 4 +… + n = n(n + 1)/2
2. Sum of squares of natural numbers: 1² + 2² + 3² + 4² + …… + n² = n(n + 1) (2n + 1)/6
3. Sum of cubes of natural numbers: 13 + 23 + 33 + 43 + …… + n3 = [n(n + 1)/2]² or (Σn)2.

Geometric Mean: If three quantities a, b, c are in G.P, then b is called the Geometric Mean between a and c, and is given by b = √
ac.
Hence, the geometric mean between 3 and 27 is √3.27 = √81 = 9.
Hence, 3, 9, and 27 are in G.P.

HARMONIC PROGRESSION (HP)

The progression a₁, a₂, a₃, …. is called an HP if 1/a₁, 1/a₂ 1/a₃, ….. is in AP. 

If a, b, c are in HP, then b is the harmonic mean between a and c.

In this case, b = 2ac/(a + c).

ARITHMETICO‐GEOMETRIC SEQUENCE

If a₁, a₂, a₃, …, a_n, … is an AP and b₁, b₂, …, b_n … is a HP, then the sequence a₁b₁, a₂b₂, a₃, b₃, …, a_nb_n, … is said to be an arithmetico-geometric sequence. 

Thus, the general form of an arithmetico geometric sequence is a, (a + d) r, (a + 2d) r², (a + 3d) r³, … 

From the symmetry we obtain that the nth term of this sequence is [a + (n – 1) d] r^{n – 1}

Ex.1. The sequence 1, 3x, 5x², 7x³, … is an arithmetico-geometric sequence whose corresponding A.P. and G.P. are 1, 3, 5, 7, … and 1, x, x², x³, … respectively. The nth term of the corresponding A.P. is 1 + (n – 1) × 2 = 2n – 1 and the nth term of the corresponding G.P. by tacking first term as unity is x^{n – 1}. So, the nth term of the given sequence is (2n – 1) x^{n – 1} 

Ex.2. The sequence 2, – 5x, 8x², –11x ², – 11x³, … is an arithmetico-geometric sequence whose corresponding A.P. and G.P. are 2, 5, 8, 11, … and 1, –x, x², – x³, … The nth term of these two sequences are 2 + (n – 1) × 3 = 3n – 1 and (– x)^{n – 1} = (–1)^{n – 1} x^{n – 1} respectively. So, the nth term of the given arithmetico-geometric sequence is (3n – 1) (– 1)^{n – 1} x^{n – 1}

ARITHMETICO‐GEOMETRIC SERIES

a, (a + d) r, (a + 2d) r², (a + 3d) r³, is an arithmetico-geometric sequence.

SUM OF N TERMS OF AN ARITHMETICO‐GEOMETRIC SEQUENCE

SUM OF AN INFINITE ARITHMETICO‐GEOMETRIC SEQUENCE