Exams Prep Master
In mathematics, a series can be defined as a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is an integral part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics, and finance. On the other hand, a sequence is defined as an enumerated collection of objects in which repetitions are allowed and order plays an important role. Here, we will be explaining the series formula with some solved examples.
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Also read: Isosceles Triangle Theorems
Popular Forms of Series
The most widely used form of series are as follows:
- Arithmetic Progression:
An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. For example, in the arithmetic progression series 5,7,9,11,13,15… the common difference between any two consecutive numbers is 2.
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- Geometric Progression:
A geometric progression is a sequence of numbers, none of which are zero, and each term after the first term is calculated by multiplication of the previous one by a fixed number. This fixed, non-zero number is called the common ratio. For example, in the geometric progression 2,6,18,54… the common ratio is 3, as 2 x 3 = 6, 6 x 3 = 18, 18 x 3 = 54
- Harmonic Progression:
A harmonic progression is a progression that is formed by the reciprocals of an arithmetic progression. For example, the series 1, 1/2, 1/3, 1/4, 1/5, 1/6… is a harmonic progression.
- Fibonacci Series:
Fibonacci numbers form a sequence called the Fibonacci sequence, in which each number is the sum of the two numbers preceding it. The series starts with 0 and 1. The sequence goes as: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…
The video below explains this:
Sequence and Series Detailed Video Explanation:
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Formula for Sum of the Terms of an Arithmetic Series
The formula used for calculating the sum of the first n terms of an arithmetic sequence is as follows:
Sn = n (a1 + a2)/2
Where,
n = number of terms
a1 = the first term
a2= the last term.
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Formula for Sum of Terms of a Geometric Series
The formula that is used for the calculation of the sum of the first n terms of a geometric series is as follows:
Sn = a1(1−rn)/1−r, r≠1
Where,
n= number of terms,
a1= the first term
r= common ratio
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Difference between Sequence and Series
The key differences between sequence and series are as follows:
| Sequence | Series |
|---|---|
| Sequence relates to the organization of terms in a particular order (i.e. related terms follow each other) and series is the summation of the elements of a sequence. | Series can also be classified as finite and infinite series. |
| In sequence, the ordering of elements is the most important. Hence, a sequence 5, 6, 7 is different from 7, 6, 5. | In a series, the order of elements is not important. Thus in case of series 5 + 6 + 7 is same as 7 + 6 + 5. |
| The elements in the sequence follow a specific pattern | The series is the sum of elements in the sequence |
| General Form: [pn]∞n=1 | General Form: Sn=∑nr=1mr |
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Points to Remember
- Sequences can be finite, or infinite, such as the sequence of all even positive integers.
- In mathematical analysis, a sequence is often denoted by letters in the form of an, bn, and cn, where the subscript n refers to the nth element of the sequence.
- In modern terminology, any (ordered) infinite sequence (a1, a2, a3…) of terms (that is, numbers, functions, or anything that can be added) defines a series, which is the operation of adding the ai-one after the other. To emphasize that there are an infinite number of terms, a series may be called an infinite series.
- An arithmetic sequence is the sequence where every term is created by adding or subtracting a definite number to the preceding number.
- A geometric sequence is the sequence where every term is obtained by multiplying or dividing a definite number with the preceding number.
- A series of numbers is considered to be in harmonic sequence if the reciprocals of all the elements of the sequence form an arithmetic sequence.
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Sample Questions
Ques. Determine the sum of the first 50 terms of the arithmetic sequence 2, 5, 8, 11, 14, 19, 22, 25, 28, 31? (2 marks)
Ans. We will have to first find the 50th term:
A50 = a1+ (n−1) d
=2+49(3)
=149
Then find the sum:
Sn = n(a1+ an)/2
S50 = 50(2 + 149)/2
=3775
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Ques. Identify S10 of the geometric series 24+12+6+..... (2 marks)
Ans. First, we will find r
r = r2/r1 = 12/24
=12
Now, we will find the sum:
S10= 24(1− (1/2)10)/1−1/2
=3069/64
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Ques. In a G.P, the second term is 12 and the sixth term is 192. Find the 11th term. (3 marks)
Ans. Given second term, ar = 12 ..(1)
Sixth term, ar5 = 192 …(2)
Dividing (2) by (1) we get ar5 / ar = 192/12
⇒ r4 = 16
⇒ r = 2
Substitute r in (1), we get a×2 = 12
⇒ a = 12/2 = 6
11the term is given by ar10 = 6×210 = 6144
Hence the required answer is 6144.
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Ques. If Sn denotes the sum of first n terms of an A.P. and [S3n − Sn−1] / [S2n −S2n−1] = 31, then the value of n is __________. (3 marks)
Ans. S3n = [3n / 2] [2a + (3n − 1) d]
Sn−1 = ([n − 1] / 2) * [2a + (n − 2) d]
⇒ S3n − Sn−1 = [1 / 2] [2a (3n − n + 1)] + [d / 2] [3n (3n − 1) − (n − 1) (n − 2)]
= [1 / 2] [2a (2n + 1) + d (8n2 − 2)]
= a (2n + 1) + d (4n2 − 1)
= (2n + 1) [a + (2n − 1) d]
S2n − S2n−1 = T2n = a +(2n − 1) d
⇒ S3n − Sn − S2n −S2n−1 = (2n + 1)
Given, ⇒ [S3n − Sn−1] / [S2n − S2n−1] = 31
⇒ n = 15
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Ques. If the sum of n terms of an A.P. is (c*n) (n – 1), where c ≠ 0 then the sum of the squares of this term is __________. (2 marks)
Ans. If tr is the rth term of the A.P., then
tr = Sr − Sr−1 = cr (r − 1) − c (r − 1) (r − 2) = c (r − 1) (r − r + 2) = 2c (r − 1)
We have, t12 + t22+……..+tn2 = 4c2 (o2 + 12 + 22… + (n − 1)2)
= [4c2] * [(n − 1) n (2n − 1)] / [6]
= [2 / 3] c2 n (n − 1) (2n − 1)
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Ques. The largest term common to the sequences 1, 11, 21, 31,…. to 100 terms and 31, 36, 41, 46,………to 100 terms is ___________. (3 marks)
Ans. 100th term of 1, 11, 21, 31, …………. is 1+ (100 – 1) 10 = 991.
100th term of 31, 36, 41, 46, ………… is 31 + (100 – 1) 5 = 526.
Let the largest common term be 526.
Then, 526 = 31 + (n – 1) 10
Or n = 50.5
But n is an integer; hence n = 50.
Hence, the largest common terms are 31 + (n – 1) 10 = 521.
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Ques. Find the sum of the series 1 + 1 / [4 * 2!] + 1 / [16 * 4!] + 1 / [64 * 6!]+………. (3 marks)
Ans. We know that ex = 1 + x / 1! + x2 / 2! + x3 / 3! + x4 / 4! + …….. ∞ ——— (i)
∴ [ex + e−x] / 2 = 1 + x2 / 2! + x3 / 3! + x4 / 4………..∞ ————-(ii)
On adding equations (i) and (ii),
we get ex + e−x = 2x + 2x2 / 2! + 2x4 / 4!+…∞
∴ [ex + e−x] / 2 = 1 + x2 / 2! + x4 / 4 + x6 / 6!+ ……………
Putting x = 1 / 2, we get
e + 1 / 2√ e = 1 + 1 / [4 * 2!] + 1 / [16 * 4!] + 1 / [64 * 6!]+……….
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Ques. If a, b, and c are in A.P., p, q, and r are in H.P., and ap, bq, and cr are in G.P., then [p / r] + [r / p] is equal to ______________. (4 marks)
Ans. p / r + r / p = [p2 + r2] / [pr]
= (p + r)2 − ([2pr] / [pr])
= [(4p2r2 / q2) − 2pr] / [pr]
[p, q, r are in H.P. Therefore, q = 2pr / [p + r]]
= [4pr / q2] − [2]
= [4b / ac] − [2]
[ap, bq, cr are in A.P ⇒ b2q2 = ac * pr]
= [(a + c)2 / ac] − [2]
[a, b, c are in A.P ⇒ 2b = a + c]
= [a / c] + [c / a]
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