Let us recall the concepts in Numbers and Sequences:
- Euclid division lemma - HCF of two numbers.
Let \(a\) and \(b (a > b)\) be any two positive integers. Then, there exist unique integers \(q\) and \(r\) such that \(a = bq + r\), \(0 \leq r < b\).
- Fundamental theorem of arithmetic - Prime factorisation method, LCM and HCF.
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.
- Congruence modulo: Two integers, \(a\) and \(b\), are said to be congruence modulo \(n\) if and only if their difference is divisible by \(n\). That is, \(a \equiv b(mod n)\).
- Arithmetic progression - Finding common difference, nth term, verify if the sequence is A.P.
1. Formula to find the \(n^{th}\) term of an A.P. is \(t_n = a + (n - 1)d\).
2. Common difference of an A.P. is \(d = t_n - t_{n - 1}\).
3. The sum of \(n\) terms in a A.P. is \(S_n = \frac{n}{2}[2a + (n - 1)d]\).
4. If the last term \(l\) (\(n^{th}\) term) is given, then \(S_{n} = \frac{n}{2}[a+l]\).
- Geometric progression - Finding common difference, nth term, verify if the sequence is G.P and applications of G.P.
1. The general form of a G.P. is \(ar^{n - 1}\).
2. The sum of \(n\) terms in a G.P. is \(S_n = \frac{a(r^n - 1)}{r - 1}\).
3. Sum of infinite terms of a G.P. is \(\frac{a}{r - 1}\), \(-1 < r < 1\).
- Special Series:
1. Sum of first \(n\) natural numbers \(= \frac{n(n + 1)}{2}\).
2. Sum of first \(n\) odd natural numbers \(= n^2\).
3. Sum of squares of first \(n\) natural numbers \(= \frac{n(n + 1)(2n + 1)}{6}\).
4. Sum of cubes of first \(n\) natural numbers \(= \left(\frac{n(n + 1)}{2} \right)^2\).