Sequence and Series: Overview, Questions, Preparation

The sequence is a collection of different objects arranged in such an orderly manner that it can be identified as the first, second, third member and so on. Sequences play an essential role in various parts of our lives. 

A series can be defined as the sum of all the terms present in a sequence. But there has to be a relationship between the terms and the sequence.

Sequence

When a sequence has a finite number of terms, it is called a Finite sequence.
E.g., the sequence of ancestors in a family is an excellent example of a finite sequence.

When a sequence has an infinite number of terms, it is called an infinite sequence.
E.g., a Fibonacci series is an excellent example of an infinite sequence.

Series

A series can be defined as infinite, or finite series based on their sequence. The series is always mentioned in short form, called sigma notation and denoted by the Greek letter ‘Sigma’.
E.g., if S1, S2, S3,…., Sn are a given sequence, then the series is defined as S1+ S2+ S3+ …. + Sn.

Formulas Related to Sequence and Series

The formulas related to arithmetic progression and geometric progression have been mentioned below.

a - first term

d - the common difference

r - the common ratio

n - position of terms

I - last term

For Arithmetic progression

Sequence- a, a+d, a+2d, …., a+(n-1)d…

Common difference- d= a2-a1

General term- an = a + (n-1)d

n^th term from the last term- an= I- (n-1)d

Sum of first n terms- s = (n/2)(2a + (n-1)d)

For geometric progression,

Sequence- a, ar, ar², ar³,…., ar^(n-1),…

Common difference- r= (ar^n-1/ar^n-2)

General term- a_n=ar^(n-1)
n^th term from the last term- an= (1/r)^(n-1))

Sum of first n terms- S= a(1-rⁿ)/(1-r); if r
S_n = a(rⁿ-1)/(r-1); if r>1

The chapter ‘Sequence and Series’, discusses the sequence and series along with Arithmetic and geometric progression. This chapter also explains the difference between arithmetic and geometric progression.

1. The first term of a GP is 1. The sum of the third and the fifth terms is 90. Find the common ratio of GP.

Solution. Let common ratio = r.

= a³ + a⁵ =90

= ar² + ar⁴ = 90

= r⁴ + r² -90 = 0

= r⁴ + 10r² -9r²- 90 =0

= r² - 9 = 0

= r = +3 or -3

2. If the sum of three numbers in AP is 24, and the product is 440, what are the numbers?

Solution. Let the three numbers be a-d, a, and a+d

Sum= (a-d)+ a + (a+1) = 24

      = 3a = 24

      = a = 8

Product= (a-d) * a * (a+d)= 440         

= d= 3

When d= +3, the numbers are 5, 8, and 11.

when d= =3, the numbers are 11, 8, and 5.

3. Find the sum of all numbers between 200 and 400, which are divisible by 7.

Solution. First terms, a= 203, Last term, an= 399, Difference= 7

an = a + (n-1)d

= 399= 203 + (n-1)*7

= n = 29

So, S₂₉ = (29/2)* (203 + 399)

S₂₉ = 8729

Q: What is a sequence?

Q: What is a series?

Q: What are the two kinds of progressions?

Q: What is the sequence used in Arithmetic progression?

Q: What is the sequence used in Geometric progression?