Series Definition, Properties With QnA

Safalta Expert Published by: Saksham Chauhan Updated Wed, 23 Mar 2022 05:43 PM IST

Highlights

A series may be defined as the addition of an unlimited number of numbers or quantities to a particular beginning number or amount in mathematics. In many areas of mathematics, including the study of finite structures, such as combinatorics for creating functions, we employ series. Calculus and its generalizations, as well as mathematical analysis, rely heavily on the understanding of series. In addition to their mathematical applications, infinite series are widely employed in quantitative sciences such as statistics, physics, computer science, and finance. Check here at Safalta.com

A series may be defined as the addition of an unlimited number of numbers or quantities to a particular beginning number or amount in mathematics. In many areas of mathematics, including the study of finite structures, such as combinatorics for creating functions, we employ series. Calculus and its generalization, as well as mathematical analysis, rely heavily on the understanding of series. In addition to its mathematical applications, infinite series are widely employed in quantitative sciences such as statistics, physics, computer science, and finance.

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Meaning of the Series
The notion of sequences may be used to define series in mathematics. Because sequence and series are two notions that are intertwined. Assume a1, a2, a3,..., an is a sequence, then the phrase a1 + a2 + a3 +,...+ an is referred to as the series associated with the sequence. Depending on whether the above sequence is finite or infinite, the series is finite or infinite. To illustrate the summation involved, series are frequently expressed in compact form, known as sigma notation, which uses the Greek symbol sigma.

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Series Number
A series can have several terms in the form of numerical values, functions, quantities, and so on. The symbolized amount, not the total itself, is indicated when the series is supplied. 2 + 4 + 6 + 8 + 10 + 12 is a six-term sequence, for example. To get the total of these numbers, we use the term "sum of a series," which refers to the number that is obtained by adding the series' terms together; the sum of the series is 42. 
 
Finite Series
A finite series is a series having a limited number of terms.
 
If a1 + a2 + a3 +... + an is an n-term series, then it is a finite series with n terms.
 
As a result, Sn stands for the sum of the series and is written as:
 
Sn = ∑ an
 
We may also specify the total of a certain number of terms. These are written as follows:
 
S1 = a1
S2 = a1 + a2
S3 = a1 + a2 + a3
S4 = a1 + a2 + a3 + a4
….
Sn = a1 + a2 + a3 + … + an
 
Questions And Answers
 
Question 1
Find the missing number from the given responses
1, 0.125, 1/27, 1/64, ?, 1/216
 
Option1: 0.025 
Option 2: 1/8 
Option 3: 1/128
Option 4: 0.008
 
Answer:
4: 0.008
Explanation:
cube of 1 ,2 ,3 ,4 ,5 ,6 in denominator.
 
Question 2
Find the missing number from the given responses
–1,0,?,8,15.,24
 
Option 1: 4
Option 2: 3
Option3:
Option4: 1
 
Answer:
2: 3
 
 
Explanation:
square of 0,1,2,3,4,5andsubstract1.
 
Question 3
Find the missing number from the given responses
0,6,24,60,?,210
 
Option 1: 117
Option 2: 119 
Option 3: 120
Option 4: 153
 
Answer:
3: 120
 
Question 4
Find the missing number from the given responses
–1,2,7,?,23,34,47
 
Option1: 13
Option 2: 14
Option 3: 12
Option4: 15
 
Answer:
2: 14
 
Explanation:
-1 → +32 → +57 → +714 → +923 → +1134 → +1347hence 14 is the right ans.
 
 
Question 5
Find the missing number from the given responses
0.01, 0.04, 0.09, ?

Option 1: 0.016 
Option 2: 1.6 
Option 3: 0.16 
Option4: 0.106
 
Answer:
3: 0.16
 
Explanation:
square of .1, .2, .3, .4
 
Question 6
Find the missing number from the given responses
6,24,12,?,18,8,24,0
 
Option1: 14 
Option 2: 18
Option 3: 16 
Option 4: 20
 
Answer:
3: 16
 
Question 7
Find the missing number from the given responses
0.15, 0.3, ?, 1.2, 2.4
Option 1: 0.6
Option 2: 0.9
Option3: 0.06 
Option 4: 4.8
 
Answer:
1: 0.6
 
Explanation:
multiply each term with 2 .
 
Question 8
Find the missing number from the given responses 3,8,18,35,?,98
 
Option 1: 61 
Option 2: 71
Option 3: 41
Option 4: 51
 
Answer:
1: 61
 
Explanation:
The sequence can be explained by looking at the difference between the numbers. Between 3 and 8 is 5..... which is a 4+1, a perfect square plus 1
Between 8 and 18 is 10... which 9+1, the next perfect square plus 1
Between 18 and 35 is 17... which 16+1, the next perfect square plus 1
So.... the between 35 and X ... will be 25 +1, which is 26 35+26=61
So, the answer is 61.
Oh, by the way, the sequence continues, so 61 + 36+1 = 98. 61 is the answer.
 
Question 9
Find the missing number from the given responses
39, 169, 299, ?

Option 1: 339 
Option 2: 429
Option 3: 519 
Option 4: 609

Answer:
2: 429
 
Explanation:
add +130 in each term .
 
Question 10
Find the missing number from the given responses.
 
2460, 3570, 4680, ?
 
Option1: 8640 
Option 2: 5670 
Option 3: 5970
 Option 4: 5790
 
Answer:
4: 5790
 
Explanation:

 
Question 11
Find the missing number from the given responses.
321, 301, ?, 181, 21
 
Option 1: 281 
Option 2: 161
Option 3: 241
Option4: 261
 
Answer:
4: 261
 
Explanation:
321-20=301
301-40=261
261-80=181 181-160=21
 
Question 12
Find the missing number from the given responses.
49, 64, ?, 100, 121
 
Option 1: 74 
Option 2: 80 
Option 3: 75 
Option 4: 81
 
Answer:
4: 81
 
Explanation:
square of 7,8,9,10,11,.
 
Question 13
Find the missing number from the given responses.
4, 5.2, 6.4, ?, 8.8
Option1:
Option 2: 8
 Option 3: 7.6
 Option 4: 8.25
Answer:
3: 7.6
 
Explanation:
add +1.2 in each term 
 
Question 14
Find the missing number from the given responses.
2, 8, 40, ?, 1680, 13440
 
Option1: 80 
Option 2: 120
Option 3: 160 
Option4: 240
 
Answer
4: 240
 
Explanation:
multiply by 4,5,6,7,8, respectively 
 
Question 15
Find the missing number from the given responses.
21, 26, 33, 42, 53, ?
Option 1: 56
Option 2: 63 
Option 3: 66 
Option 4: 69
 
Answer
3: 66
 
Explanation:
add +5 , +7, +9, +11, +13 respectively.
 
Question 16
Find the missing number from the given responses.
6,?,15,27,51,99
 
Option 1: 10 
Option 2: 9
Option 3 :11
Option4: 12
 
Answer:
2: 9
 
Explanation:
6 × 2 = 12 - 3 = 99 × 2 = 18 - 3 = 1515 × 2 = 30 - 3 = 2727 × 2 = 54 - 3 = 5151 × 2 = 101 - 3 = 99
 
 
Question 17
Find the missing number from the given responses. 5, 13, 29, 61, 125, ?
 
Option 1: 253 
Option 2: 196 
Option 3: 245 
Option 4: 145
Answer: 1: 253
Explanation:
multiply by 2 and add 3 in each term .
 
 
Question 18
Find the missing number from the given responses 242, 393, 4164, ?
 
Option1: 5525
Option 2: 5255 
Option 3: 5235 
Option 4: 5325
 
Answer:
2: 5255
 
Explanation:
242, 393, 4164, ?
Ist series 2, 3, 4, 5 IIndseries4,4+5=9,9+7=16,16+9=25 ∴ the answer is 5255.
 
 
Question 19
Find the missing number from the given responses.
3,5,8,13,21,34,55,?,
 
Option 1: 34
Option 2: 55
Option 3: 21 
Option 4: 89
 
Answer:
4: 89
 
Explanation:
3+5=8 5+8=13 8+13=21 13+21=34 21+34=55 34+55=89
 
Question 20
Find the missing number from the given responses.
313, 623, 933, 1243, ?
Option 1: 1863 
Option 2: 2173
Option 3: 1553 
Option 4: 2483
 
Answer:
3: 1553
 
Explanation:
add 310 in each term 
 
 
 

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