Title XI - maths - chapter 9 - SEQUENCE_SERIES (140-173).pdf ✅

140 NARAYANAGROUP SEQUENCE & SERIES JEE-MAIN-SR-MATHS VOL-I SEQUENCE AND SERIES Sequence: A set of numbers is arranged in a definite order according to some definite rule is called a sequence. e.g. 2, 4, 6, 8, ....., is a sequence  A sequence is a function whose domain is a set of natural numbers. If the range of a sequence is a subset of real numbers (or complex numbers), then it is called a real sequence (or complex sequence) Series: The sum of the terms of a sequence is called a series.  If 1 2 3 a a a , , ,...... is a sequence, then the expression 1 2 3 a a a   ...... is a series  A series is called finite series, if it has finite number of terms. Otherwise it is called infinite series. e.g .i) 1+3+5+ ..........+21 is a finite series. ii) 2+4+6+8+....... is an infinite series.  Sequences following specific patterns are called progressions. Arithmetic progression (A.P):-  A sequence is called an arithmetic progression, if the difference between any two consecutive terms is the same.  A.P is of the form a a d a d a d , , 2 , 3    ..... where a is 1st term and d is common difference General term of an A.P:  Let ‘a’ be the first term and ‘d’ be common difference of an A.P, then its genaral term (or)   th n term is 1 T a n d n     If ‘l’ be the last term and ‘d’ be common difference of an A.P, then th m term from the end ' Tm = l-(m-1)d  th m term from the end = th (n-m+1) term from the beginning. W.E-1: Find the first negative term of the sequence 1 1 3 20,19 ,18 ,17 ,...... 4 2 4 Sol: The given sequence is an A.P in which first term a=20 and common difference d=-3/4. Let the nth term of the given A.P. be the first negative term. Then, an  0          a n d n ( 1) 0 20 ( 1)( 3/ 4) 0 83 3 0 83 3 0 3 83 4 4 n         n n 2 27 3   n   n 28 thus, 28th term of the given sequence is the first negative term. W.E-2: If 100 times the 100th term of an A.P with non-zero common difference equals the 50 times of 50th term, then find 150th term of this A.P. (AIEEE 2012) Sol: 100 50 100 50 T T  ; 100(a+99d)=50(a+49d) 2a+198d=a+49d ; a+149d=0 T a d 150    149 0 Sum to n terms of an A.P:   2 n n S a l   2 1   2 n      a n d   where a  first term, l  last term d  common difference  If the sum of n terms of a sequence Sn is given, then its th n term Tn can be determined by T S S n n n   1 W.E-3: How many terms are to be added to make the sum 52 in the series (-8)+(-6)+ (-4)+....? Sol: 52 2 8 1 2 52     2 n n S n              n n 2 18 104       n n n  9 52 13  SYNOPSIS
NARAYANAGROUP 141 JEE-MAIN-SR-MATHS VOL-I SEQUENCE & SERIES W.E-4: Let 1 2 , ,....., a a an be the terms of an A.P. If 1 2 1 2 ..... ..... p q a a a a a a       = 2 2 p q , p q  then find 6 12 a a . Sol:         2 1 1 2 1 1 1 6 21 1 6 21 2 1 2 1 2 2 1 2 1 2 1 2 , 11, 41 1 2 11 41 p a p d p p a p d q q q a q d a q d p a d p a For p q q q a a d a a                                                  Properties of A.P:-  a b c , , are in AP    2b a c  In a finite A.P, the sum of the terms equidistant from the begining and the end is always same and is equal to the sum of the first and last term 2 1 3 2 4 3 1 . ., n n n n i e a a a a a a a a            a a a a 1 2 3     ...... n      middleterm , if isodd sumof twomiddleterms , if iseven 2 n n n n          If 1 2 3 , , ........... a a a an are in A.P then a) 1 3 2 1 , ,..... , , a a a a a n n are in A.P b) 1 2 3 , , ; ....... a a a a         n are in A.P (where R ) c) 1 2 3 , , ; .......     a a a an are in A.P (where   R {0})  th p term of an A.P. is ‘q’ and th q term is ‘p’, then Tp q  0  If th m term of an A.P. is ‘n’ and th n term is ‘m’ then th p term is ‘m+n-p’  If S q p  and S p q  for an A.P., then S p q p q      W.E-5:If     1 3 3 1 1, log 3 2 ,log 4.3 1 2 x x   are in A.P, then find x. Sol:     1 3 3 1 1, log 3 2 ,log 4.3 1 2 x x   are in A.P     1 3 3 log 3 2 1 log 4.3 1      x x     1 3 3 3 log 3 2 log 3 log 4.3 1      x x     1 3 3 log 3 2 log 3 4.3 1     x x     1 3 2 3 4.3 1     x x 3.3 2 12.3 3     x x   3 2 12 3, 3x t where t t      2 2         3 2 12 3 12 5 3 0 t t t t t     4 3 3 1 0 t t     3 1 3 , 3 3 0 4 3 4 x x t        3 3 3 log 1 log 4 4 x           Selection of terms in an A.P: W.E-6: If the sum of four numbers in A.P is 24 and the sum of their squares is 164 then find those numbers. Sol: (a-3d)+(a-d)+(a+d)+(a+3d)=24     4 24 6 a a         2 2 2 2 a d a d a d a d         3 3 164     2 2 2 2      2 9 2 164 a d a d 2 2 2          a d d d 5 41 36 5 41 1 required numbers are 3,5,7,9