Sequence and Series Questions and Answers

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Sequence and Series of Quantitative Aptitude

Sequence and Series of Quantitative Aptitude

Arithmetic Progression: If the difference between any two consecutive terms of a sequence is same  then the sequence  is said to be arithmetic progression. It is denoted by A.P.

Example :  1,5,9,13,17,……………

-2,-5,-8,-11,-14,…………..

 

nth term or last term of an A.P. is given by the formula

an= a + (n-1) d

Where   a = first term of an A.P.,

              d = common difference

              n = number of terms

 

Arithmetic mean between any two given quantities a and b = (a+b)/2

 

Sum of first n terms of an A.P.

sn= n2 2a + (n-1)d or

sn = n2 2a +l

Where    l = the last term of an A.P.

 

Geometric progression: If the ratio of any two consecutive terms of a sequence is same then the sequence is said to be in geometric progression. It is denoted by G.P

Example 1,½,¼,⅛,1/16,……………………….

3, -9/4, 27/16, -81/16,…………….

 

nth term of a G.P. is given by the formula

.an= arn-1

Where      a = first term

                R = common ratio

 

Sum of n terms of G.P.
sn= a (rn-1)/r-1 ,                    if r 1  and

sn= an ,                                  if r = 1

 

Sum of infinite terms  of a G.P. in case of  -1 < r <1

sn= a/(1-r)

 

Geometric mean between any two given quantities a and b = ab

                                                                                                                                  

Harmonic progression: a sequence a,b,c,d,…………is said to be in harmonic progression if 1/a,1/b,1/c,1/d,………………  is in A.P.

Example: ½,¼,⅙,……………..

 

Harmonic mean between any two given quantities a and b = 2ab/(a+b)

Following Question Prepared by Best Bank Coaching in Noida

Sum of special type of series:

  • The sum of first n natural numbers i.e. 1+2+3+………+n = n (n+1)/2
  • The sum of squares of first n natural numbers i.e.12+22+….+n2 = n(n+1)(2n+1)/6
  • The sum of cubes of first  n natural numbers i.e. 13+23+…….+n3=n2(n+1)2/4
  • The sum of first n odd natural numbers  =  n2
  • The sum of first n even natural numbers = n2+ n
  • The sum of squares of even natural numbers upto n2= n(n+1)(n+2)/6
  • The sum of squares of odd natural numbers upto n2= n(n+1)(n+2)/6

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Sequence and Series of Quantitative Aptitude WORKED -OUT PROBLEMS

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  1. In the sequence of numbers 0, 7, 26, 63, ….., 215, 342 the missing term is

      Answer: 124

                     0 = 13-1

                     7 = 23-1

                     26 = 33-1

                     63 = 43-1

       So missing number is = 53-1 = 124

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   2) Which number in the sequence 41, 43, 47, 53, 61, 71, 73, 81, is wrongly written ?

   Answer:  81

   In the sequence, all numbers are prime except 81

 

   3) the sum (101 + 102 + 103 + …………..+ 200) is equal to

    Answer : 15050

     This is an A.P. with first term (a) = 101 , last term (l) = 200 and common difference = 1

    an= a + (n-1)d

     200 = 101 + n-1

     n= 200-101+1 = 100

     Sum = n/2a+l

             = 100/2 101+200

             = 50*301

            = 15050

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    4) If the 4th term of an arithmetic progression is 14 and the 12th term is 70, then the first

        term is

    Answer: -7

    4th term = a4= a + (4-1)d

                          = a + 3d = 14                                                                     (1)

    12th term = a12= a + (12-1)d

                             = a + 11d = 70                                                                 (2)

    Subtract (1) from (2) , we get

     8d = 56

      d = 7

    Put the value of d in (1)

    a + 21 = 14

    a = -7

 

    5) Find the value of  1 – 1/20 + 1/202– 1/203+……………………. Correct to 5  places of

        Decimal is

    Answer: 0.95238

     This is an infinite  geometric progression with  a = 1 and  common ratio = -1/20

Sum = a/(1-r)

        = 1/(1+1/20)                                                                                              

        = 1/(21/20)

       = 20/21

       = 0.9523809

       = 0.95238 (upto 5 decimal)

 

6) Find the sum of the first n terms of the series

   5 + 55 + 555 + 5555 +…………………

Answer: 5081(10n-1) –59n

5 + 55 + 555 + …………. upto n terms

5(1 + 11 + 111 + …………upto n terms )

59 (9 + 99 + 999 + ……….upto n terms)

59(10-1 + 100-1 + 1000-1 +…………..upto n terms)

59 (10 + 100 + 1000 +…………..upto n terms – n)

5910(10n-1)10-159n

5081(10n-1) –59n

 

7) When simplified, the sum  1/2  + 1/6  + 1/12 + 1/20 + 1/30 + ………. + 1/n(n+1) is equal to

Answer: n/(n+1)

1/2  + 1/6  + 1/12 + 1/20 + 1/30 + ………. + 1/n(n+1)

= 1/(1*2) + 1/(2*3) + 1/(3*4) + 1/(4*5) +…………..+ 1/n(n+1)

= 1/1 – 1/2 + 1/2- 1/3  + 1/3 – 1/4 + 1/4 – 1/5 +………….+1/n – 1/ (n+1)

= 1-1/(n+1)

= (n+1-1)/(n+1)

= n/(n+1)

 

8) Find the sum of the following series
11*4+ 14*7+17*10+110*13+ 113*16

Answer: 5/16

11*4+ 14*7+17*10+110*13+ 113*16

= 1331*4+ 34*7+37*10+310*13+ 313*16

=13(1 – 1/4  + 1/4  -1/7  + 1/7 – 1/10 + 1/10 – 1/13 + 1/13 – 1/16 )

= 13(1 – 1/16)

= 13(15/16)

= 5/16

                                                                                                                            

9) 1 + (3 +1 ) (32+ 1)  (34+ 1)  (38+ 1)  (316+ 1)  (332+1) is equal to  

Answer: (364+ 1)/2

1 + (3 +1 ) (32+ 1)  (34+ 1)  (38+ 1)  (316+ 1)  (332+1)

 

=1 + (3 – 1) (3 +1 ) (32+ 1)  (34+ 1)  (38+ 1)  (316+ 1)  (332+1)3-1

 

= 1 +  (3 – 1)(3 +1 ) (32+ 1)  (34+ 1)  (38+ 1)  (316+ 1)  (332+1)/2

= 1+(32– 1)  (32+ 1)  (34+ 1)  (38+ 1)  (316+ 1)  (332+1)/2

= 1+(34– 1) (34+ 1)  (38+ 1)  (316+ 1)  (332+1)/2

= 1 + (38– 1)  (38+ 1)  (316+ 1)  (332+1)/2

= 1 +(316– 1)  (316+ 1)  (332+1)/2

= 1 + (332-1) (332+1)/2

=1 + (364-1)/2

=( 2 + 364– 1 )/2

= (364+ 1)/2

 

10) Given that 13+ 23+ 33+ ………….+ 103= 3025, the value of 23+ 43+ 63+………….+ 203is equal to

Answer: 24200

23+ 43+ 63+………….+ 203

= 8(13+ 23+ 33+ ………….+ 103)

= 8*3025

= 24200

 

                                                                                                                        

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