These Class 9 Maths Notes and Chapter 8 Predicting What Comes Next Exploring Sequences and Progressions Class 9 Ganita Manjari Notes are designed according to the latest CBSE syllabus.
Class 9 Maths Chapter 8 Predicting What Comes Next Exploring Sequences and Progressions Notes
Class 9 Maths Ganita Manjari Chapter 8 Notes
Class 9 Predicting What Comes Next Exploring Sequences and Progressions Notes
→ Introduction: We come across patterns in our day to day lives on almost daily basis. In nature, in art, in music, in finance and in many others aspect of our lives we see patterns. Patterns help us make sense of the world and predict what comes next.
Sequences are special kinds of patterns formed by numbers or other objects arranged in a particular order. These sequences are important in solving many real-ife problems.
In this chapter, we shall explore patterns in sequences of numbers. We will then find rules to help us predict more numbers of the sequence.
We have already seen some of the number sequences in our previous classes, like 1, 2, 3, 4, 5, 6, … (Natural Numbers), 1, 3, 5, 7, 9, 11, … (odd numbers), 1, 3, 6, 10, 15, 21, 28, … (triangular numbers), 1, 4, 9, 16, 25, 36, 49, 64, … (square numbers), 2, 4, 6, 8, 10, 12, … (even numbers), etc.
→ Sequence:
- Sequence is an ordered list of numbers where each number is a term of the sequence. Thus in the sequence of cube numbers, 1 is the first term, 8 is the second term, 27 is the third term, 64 is the fourth term, 1000 is the tenth term.
- Sequence may be finite or infinite. The sequences mentioned in the introduction are all infinite sequences, but if we write 3, 6, 9, 12, 15 as a sequence then it is a finite sequence having only 5 terms.
→ Explicit Rule for a Sequence:
Using the notation tn (or Sn or un) we can write an explicit rule for the term in the nth position of a sequence, that is, the nth term. An explicit formula uses the term’s position number, n, to calculate its value. For example, let us consider the expression tn = 5n – 2. This states that the nth term of the sequence is given by the rule 5n – 2. When we substitute n- 1, 2, 3, 4,… in the expression 5n – 2, we get t1 = 5 × 1 — 2 = 3, t2 = 5 × 2 – 2 = 8, t3 = 5 × 3 – 2 = 13, t4 = 5 × 4 – 2 = 18, etc.
Thus, tn = 5n – 2 is the explicit rule for the nth term of the sequence.
→ Recursive Rule for a sequence:
The way of describing a sequence by relating terms to previous terms is known as a recursive rule or recursive formula. For example, we can define u1 = 4, un = 4un-1 terms of this sequence as u1 = 4,
u2 = 4u2-1 + 3 = 4u1 + 3 = 4 × 4 + 3 = 19.
u3 = 4u3-1 + 3 = 4u2 + 3 = 4 × 19 + 3 = 79,
u4 = 4u4-1 + 3 = 4u3 + 3 = 4 × 79 + 3 = 319.
Thus, the first four terms of the sequence are 4, 19, 79, 319.
→ Virahanka-Fibonacci sequence:
A recursive rule or formula does not only have to involve the previous term- it could involve the previous two terms or even more.
The most famous example of such a sequence is a1, a2, a3…, where a1 = 1, a2 = 1 and
an = an-1 + an-2 for n ≥ 3. We can write a3 = a3-1 + a3-2 = a2 + a1 = 1 + 1 = 2,
a4 = a4-1 + a4-2 = a3 + a2 = 2 + 1 = 3, a5 = a5-1 + a5-2 = a4 + a3 = 3 + 2 = 5,
a6 = a6-1 + a6-2 = a5+ a4 = 5 + 3 = 8, ……..
So, we can list first few terms of the Virahanka – Fibonacci sequence as: 1, 1, 2, 3, 5, 8, ….
→ Arithmetic Progression (AP):
An arithmetic progression (AP) can be described as
a, a + d, a + 2d, a + 3d,…, a + (n – 1) x d, where ‘a’ is the first term and ‘d’ is the common difference.
Thus, tn = a + (n – 1) × d is an expression for the nth term of any arithmetic progression, for some fixed values of a and d.
→ Sum of the First n Natural Numbers:
Let S = 1 + 2 + 3 + 4 + …. + n = \(\frac{n(n+1)}{2}\)
→ Geometric Progression (GP):
A geometric progression (GP) can be described as
a, ar, ar2, ar3,…, arn-1 where ‘a’ is the first term and ‘r’ is the common ratio.
Thus, tn = a × rn-1 is an expression for the nth term of any geometric progression, for some fixed values of a and r.