Numbers and Sequence:
The chapter "Numbers and Sequence" carrying the weightage of \(18\) marks in the board examination. It covers the concepts like Euclid division lemma, Fundamental theorem of arithmetic, Modular arithmetic, Sequences, Arithmetic progression, Sum to n terms of an A.P, Geometric progression, Geometric Series and Special series.
| Total marks \(18\) |
Total marks \(= 18\)
|
To prepare well for the board examination, it is necessary to understand the following concepts clearly.
- Euclid division lemma - HCF of two numbers.
- Fundamental theorem of arithmetic - Prime factorisation method, LCM and HCF.
- Arithmetic progression - Finding common difference, nth term, verify if the sequence is A.P.
- Geometric progression - Finding common difference, nth term, verify if the sequence is G.P and applications of G.P.
| Important Concept (Learning Outcomes) | Expected Question Type | Concept dealt with |
| Sequence | Sec A | Sequence |
| Arithmetic progression and Geometric progression | Sec A, Sec B and Sec C | |
| Sum of the series | Sec C | Sum to n terms of the series |
Let us recall the concepts in Numbers and Sequences:
1. Let \(a\) and \(b (a > b)\) be any two positive integers. Then, there exist unique integers \(q\) and \(r\) such that \(a = bq + r\), \(0 \leq r < b\).
2. Fundamental theorem of arithmetic: Every composite number can be expressed (factorised) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.
3. Two integers, \(a\) and \(b\), are said to be congruence modulo \(n\) if and only if their difference is divisible by \(n\). That is, \(a \equiv b(mod n)\).
4. Formula to find the \(n^{th}\) term of an A.P is \(t_n = a + (n - 1)d\).
5. Common difference of an A.P is \(d = t_n - t_{n - 1}\).
6. Sum of series to \(n\) terms is \(S_n = \frac{n}{2}[2a + (n - 1)d]\).
7. The general form of a G.P is \(ar^{n - 1}\).
8. The sum of \(n\) terms in a G.P is \(S_n = \frac{a(r^n - 1)}{r - 1}\).
9. Sum of infinite terms in a series is \(\frac{a}{r - 1}\), \(-1 < r < 1\).
10. Sum of first \(n\) natural numbers \(= \frac{n(n + 1)}{2}\).
11. Sum of first \(n\) odd natural numbers \(= n^2\).
12. Sum of squares of first \(n\) natural numbers \(= \frac{n(n + 1)(2n + 1)}{6}\).
13. Sum of cubes of first \(n\) natural numbers \(= \left(\frac{n(n + 1)}{2} \right)^2\).