SEQUENCE & SERIES

Introduction

 In Arithmetic one of the initial topics is known as Series and sequence. Sequence and series are part of Quantitative aptitude. A function is remarked as a sequence and a group of integers that are positive is identical to the sequence’s domain. 

A table of events that can be objects can be a sequence. The sequence can be 1, 5,9,13, because the difference of every term is 4, and adding 4 with the first term gives the result 1+4 = 5 and similar to the next number. The result of the sequence has been applied through the rule of an = a + (n – 1) d, in a means nth term, a means first term and d means the difference between the number. Instances can be 1, 2, 3 … Find the nth term? The solution is an = a + (n-1)*d = 1+ (n-1) * 1 = n. The term goes to the infinitive and the answer is infinitive.

A series consists of a sum of sequences. The series and sequence have a particular relationship. In order to group the series and sequences are much related, however, a sequence has initial differences. The digit of terms = the height of the sequence. A group of digits is a sequence. The rules that have been followed are a1, a2, a3, a4, a1 means the first term, a2 means 2nd term,….. The terms can be denoted 1, 2,3,…..A digit of terms is a sequence that can be infinite or finite. Whether a sequence can be a1, a2, a3, a4,… Then the series can be SN = a1+a2+a3 + + an, SN is known as the sum of all terms. The finite sequence can be a finite series and an infinite series can be an infinite sequence that is not always true. The result depends on both series and sequence that is finite or infinite.

 The character of the series and sequence

They are mainly found in kinds of series and sequences such as Arithmetic sequences, Geometric series, Harmonic series, Harmonic series, and Fibonacci series.

Arithmetic sequences

A sequence is known as arithmetic whether in order to precede digit,  add, or subtract create all terms that happen in a particular digit.

Geometric sequence

A number that can multiply or divide to obtain all terms sequence and the digit is in preceding terms can be called a geometric.

Harmonic sequence

An arithmetic sequence can form every substance of a sequence and that is reciprocal called harmonic sequences.

Fibonacci digits

An exciting sequence is found as a Fibonacci sequence. This sequence can consist of adding elements. Zero and one are the beginning of Fibonacci numbers. F1 = 1, F0 = 0 and F1 = F n-1 + F n-2.

Formulas of sequence and series

The formula of the sequence is a+d, a+2d, a+(n-1)d, where a is called the first term of the sequence, d is known as the common difference between the two numbers, and n is called the number of terms. 

The ratio is a common difference. A common difference is known as the between the preceding term and successive term and that is d = a2 – a1.

The nth term that is known as simple term and the formula of nth term is an = a + (n-1)d

The final term of nth term is an = l – (n-1)d 

The first nth terms sum is  sn = n/2(2a + (n-1)d)

Above terms are all in Arithmetic progression.

Geometric progression of a sequence is a, ar, ar2,….,ar(n-1),…

Common difference in geometric progression is r = ar(n-1)/ar(n-2)

nth term in geometric progression is an = ar(n-1)

The final term nth term is an = 1/r(n-1)

The initial nth terms sum is sn = a(1 – rn)/(1 – r) if r < 1

sn = a(rn -1)/(r – 1) if r > 1

Where a defines the terms that are first, d defines common difference, r means common ratio, n defines terms position and small l means the final term.

Differences between series and sequence

A pattern is followed through a group of elements that is called a sequence. On the other hand, a pattern is followed through the sum of elements that are known as a series. In sequence, there is an essential order of elements. On the other hand, in series, the elements that are in order are not very necessary. 5, 6, 7, 8, 9 is a finite sequence. On the other hand, 5+6+7+8+9 is a finite series. 1, 2, 3 is said to be an infinite series. On the other hand, 1+2+3 is said to be an infinite series. 

Conclusion

In a specific order that can arrange the number is a sequence. Preparing any exam series and series is the most vital topic that has been discussed. Predict and evaluate are helped by the use of series and sequence. Moreover, a result can be monitored through the sequence and series. Many decisions can help through them.