Numbers and Sequences: Euclid’s division lemma, Fundamental theorem of arithmetic, Arithmetic Progression, Geometric Progression, Special Series

**Points to Remember**

If *a* and *b* are two positive integers then there exist unique integers *q* and *r* such that *a* = *bq* + *r*, 0 ≤ *r* < |*b|*

Every
composite number can be expressed as a product of primes and this factorization
is unique except for the order in which the prime factors occur.

**Arithmetic Progression**

(i) Arithmetic
Progression is *a* , *a* + *d* , *a* + 2*d* ,
*a* + 3*d*,….n* ^{th }*term is given by

(ii) Sum
to first *n* terms of an A.P. is

(iii) If
the last term *l* (*n ^{th}* term is
given, then

(i) Geometric
Progression is *a* , *ar*
, *ar*^{2}
,….,*ar ^{n}*

(ii) Sum
to first *n* terms of an G.P. is

(iii) Suppose
*r* =1 then *S* * _{n}* =

(iv) Sum
to infinite terms of a G.P. *a* + *ar* + *ar*^{2} +
is Where –1< *r* < 1

(i) The
sum of first *n* natural numbers 1 + 2 + 3 + + *n* =

(ii) The
sum of squares of first *n* natural
numbers

(iii) The
sum of cubes of first *n* natural
numbers

(iv) The
sum of first *n* odd natural numbers 1 + 3 + 5 + .... .. + (2*n* − 1) = *n*^{2}

Tags : Numbers and Sequences | Mathematics Numbers and Sequences | Mathematics

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