# Sequence and Series

## Sequence and Series Synopsis

**Synopsis**

**Introduction to Sequence**

• A**sequence**is an ordered list of numbers and has the same meaning as in conversational English.

• A sequence is denoted by <a_{n}> (n ≥ 1) = a_{1}, a_{2}, a_{3}, …….a_{n}

• The various numbers occurring in a sequence are called its terms.

• The n^{th}term <a_{n}> is called the general term of the sequence.

• The expression obtained by adding all the terms of a sequence is called the*series*associated with the given sequence.**Finite and Infinite sequences**

• A sequence containing finite number of terms is called a*finite sequence*. A finite sequence has a last term

• A sequence which is not a finite sequence, i.e. containing infinite number of terms is called an*infinite sequence*. There is no last term in an infinite sequence.

**Fibonacci sequence**

Fibonacci sequence 1, 1, 2, 3, 5, 8,… is generated by the recurrence relation given by

a_{1}= a_{2}= 1

a_{3}= a_{1}+ a_{2}……

a_{n}= a_{n-2}+ a_{n-1}, n > 2

**Arithmetic Progression**

• A sequence is said to be an*arithmetic progression*or A.P., if every term differs from the preceding term by a constant number.

• For example, sequence a_{1}, a_{2}, a_{3},… a_{n}is called an arithmetic sequence or an A.P. If a_{n+1}= a_{n}+ d, for all n ε N, where ‘d’ is a constant called the common difference of the A.P.**• General term of an A.P.**

i. The n^{th}term or general term of an A.P. is a_{n}= a + (n – 1)d, where a is the first term and d is the common difference.

ii. General term of an A.P., given its last term ‘λ’ is λ – (n – 1)d.

iii. General form of an A.P. is a, a + d, a + 2d,… a + (n - 1)d. ‘a’ is called the**first term**of the A.P. and ‘d’ is called the**common difference**of the A.P. ‘d’ can be any real number.

iv. Let ‘a’ be the first term and ‘d’ be the common difference of an A.P. with 'm' terms. Then the n^{th}term from the end is the (m – n + 1)^{th}term from the beginning.

•**Sum of n terms of an A.P.**

i. Let a, a + d, a + 2d, …, a + (n – 1)d be an A.P.

Then, where 𝓁 = a + (n – 1) d

ii. A sequence is an A.P. if and only if the sum of its n terms is an expression of the form Xn^{2}+ Yn, where X and Y are constants.

•**Arithmetic Mean**

i. ‘A’ is the**arithmetic mean**of two numbers ‘a’ and ‘b’ if a, A and b form an arithmetic progression and it is given by

ii. Let A_{1}, A_{2}, A_{3}, … A_{n}be n numbers between a and b such that a, A_{1}, A_{2}, A_{3}, …An, b is an A.P.

These n numbers between a and b are as follows:

• Nature of A.P. depending upon the common difference d

i. If d > 0 then the A.P. is increasing.

ii. If d < 0 then the A.P. is decreasing.

iii. If d = 0 then the A.P. is constant.

•**Selection of Terms**i. If the number of terms are three with common difference ’d’, then the three terms can be taken as a – d, a, a + d.

ii. If the number of terms are four with common difference '2d', then the terms can be taken as a – 3d, a – d, a + d, a + 3d.

iii. If the number of terms are five with common difference 'd', then the terms can be taken as a – 2d, a – d, a + d, a + 2d.

iv. If the number of terms are six with common difference '2d', then the terms can be taken asa – 5d, a – 3d, a – d, a + d, a + 3d, a + 5d.

v. If the terms of an A.P. are selected at regular intervals, then the selected terms form an A.P.

Let a_{n}, a_{n+1}, a_{n+2}be the consecutive terms of an A.P., then 2a_{n+1}= a_{n}+ a_{n+2}**Properties of Arithmetic Progression**

• If a constant is added to each term of an A.P., the resulting sequence is also an A.P.

• If a constant is subtracted from each term of an A.P., then the resulting sequence is also an A.P.

• If each term of an A.P. is multiplied by a constant, then the resulting sequence is also an A.P.

• If each term of an A.P. is divided by a non–zero constant, then the resulting sequence is also an A.P.**Geometric Progression**• A sequence is said to be a**geometric progression**or G.P., if the ratio of any of its term to its preceding term is the same throughout. Constant ratio is common ratio denoted by ‘r’.

•**General term of a G.P.**i. General form of a G.P. is a, ar, ar^{2}, ar^{3},... where ‘a’ is the first term and ‘r’ is the constant ratio, ‘r’ can take any non-zero real number.

ii. General term of a G.P. is ar^{n-1}where ‘a’ is the first term and r is the common ratio.

• Sum of a G.P.

i. Sum to first n terms of a G.P. is S_{n}= a + ar + ar² + … + ar^{n-1}

♦ If r = 1, S_{n}= a + a + a + … + a (n terms) = na

♦ If r < 1, S_{n}=

♦ If r > 1, S_{n}=

ii. The sum of an infinite G.P. with first term ‘a’ and common ratio r ( –1 < r < 1) is given by

•**Geometric Mean**i. If three numbers are in G.P., then the middle term is called the**geometric mean**of the other two.

ii. The**geometric mean**(G.M.) of any two positive numbers ‘a’ and ‘b’ is given by

iii. Let G_{1}, G_{2},… G_{n}be n numbers between positive numbers a and b such that a, G_{1}, G_{2}, G_{3},… G_{n}, b is a G.P.

•**Selection of Terms**

i. If the number of terms of a G.P. is 3, with the common ratio r, then the selection of terms can be

ii. If the number of terms of a G.P. is 4, with the common ratio , then the selection of terms can be

iii. If the number of terms of a G.P. is 5, with the common ratio r, then the selection of terms can be**Properties of Geometric Progression**

• A sequence in geometric progression will remain in geometric progression if each of its terms is multiplied by a non-zero constant.

• A sequence obtained by multiplying two G.Ps term by term results in a G.P. with the common ratio is equal to the product of the common ratios of the two G.Ps.

• The reciprocals of the terms of a given G.P. form a G.P. with common ratio .

• If each term of a G.P. be raised to the same power, the resulting sequence also forms a G.P.**Relation between A.M. and G.M.**

• If A and G be the A.M. and G.M. of two given positive real numbers ‘a’ and ‘b’ respectively, then

A ≥ G where .

Hence, the quadratic equation having a and b as its roots is x^{2}– 2Ax + G^{2}= 0

• Let A and G be A.M. and G.M. of two given positive real numbers ‘a’ and ‘b’, respectively.

Hence, the given numbers are: .

• If A.M and G.M of two numbers are in the ratio of m : n, then the given numbers are in the ratio**Harmonic Progression**

• A sequence is said to be a**harmonic progression**or H.P., if the reciprocal of the numbers form an A.P.**• General (n**^{th}) term of an H.P.

i. The n^{th}term of an H.P. is the reciprocal of the n^{th}term of the A.P. formed by the reciprocals of the terms of the H.P.

ii. If the given H.P. is then its n^{th}term is

•**Harmonic Mean**

i. If a, H, b are three quantities in H.P., then**H**is said to be the**Harmonic Mean**between a and b.

ii. a, H, b are in H.P. are in A.P.

iii. If a_{1}, a_{2}, a_{3}, …, a_{n}are n non-zero numbers in H.P., then their harmonic mean will be**Relation between A.P., G.P. and H.P.**

• Three numbers a, b, c will be in A.P., G.P. or H.P. according as

• If A, G and H are respectively the Arithmetic, Geometric and Harmonic Means between any two unequal positive numbers, then

i. A, G, H are in G.P.

ii. A > G > H**Arithmetico-Geometric Series**

• A series in which each term is the product of corresponding terms of an A.P. and a G.P. is called an*Arithmetico-Geometric series*.

•**General term of an Arithmetico-Geometric series**

i. The general or standard form of such a series is

a + (a + d) + (a + 2d)r^{2}+ (a + 3d)r^{3}+ … + {a + (n – 1)d}r^{n – 1}

Here, each term if formed by multiplying the corresponding terms of the two series:

A.P: a + (a + d) + (a + 2d) + … (a + (n – 1)d) + …

G.P: 1 + r + r^{2}+ … + r^{n – 1}+ …

ii. N^{th}term of an Arithmetico-Geometric series is given by

T_{n}= {a + (n – 1)d}r^{n – 1}

•**Sum of an Arithmetico-Geometric series**

i. The sum of the n terms of the series is given by

ii. Sum of an infinite Arithmetico-Geometric series given by

**Special Series**

• The sum of first ‘n’ natural numbers is

• Sum of squares of the first n natural numbers is

• Sum of cubes of first n natural numbers

• Sum of powers of 4 of first n natural numbers

• Consider the series a_{1}+ a_{2}+ a_{3}+ a_{4}+ ... + a_{n}+ ...

If the differences a_{2}- a_{1}, a_{3}- a_{2}, a_{4}- a_{3}, ... are in AP, then the n^{th}term is given by

a_{n}= an^{2}+ bn + c, where a, b, c are constant.

• Consider the series a_{1}+ a_{2}+ a_{3}+ a_{4}+ ... + a_{n}+ ...

If the differences a_{2}- a_{1}, a_{3}- a_{2}, a_{4}- a_{3}, ... are in G.P., then the n^{th}term is given by

a_{n}= ar^{n-1}+ bn + c, where a, b, c are constants.

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