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Sequences and Series Test - 11

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Sequences and Series Test - 11
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  • Question 1
    1 / -0
    If the sum of the first n natural numbers is 1/5 times the sum of the their squares, then the value of n is
    Solution
     n(n+1) 2=15(n(n+1)(2n+1) 6 )15=2n+1n=7\displaystyle \frac { n\left( n+1 \right)  }{ 2 } =\frac { 1 }{ 5 } \left( \frac { n\left( n+1 \right) \left( 2n+1 \right)  }{ 6 }  \right) \\ \Rightarrow 15=2n+1\Rightarrow n=7
  • Question 2
    1 / -0
    Find out next term of the series   2, 7, 28, 63, 126, ....
    Solution
    Given the series' terms can be written as 

     13+1\displaystyle1^{3}+1231\displaystyle2^{3}-133+1\displaystyle3^{3}+1431\displaystyle4^{3}-153+1\displaystyle5^{3}+1631\displaystyle 6^{3}-1 etc. 

    Hence the next number is 631=2161=2156^3-1=216 - 1 = 215
  • Question 3
    1 / -0
    The spread sheet on the right contains 20 cells. A cell In a spread sheet can be identified fIrst by the column letter and then by the row number. For example,. the number 10 is found in cell C4. If the number in cell A3 =B4-3(E2 +D4) then which of the following must be the number in cell E2?

    Solution
    Its given
    A3=B4-3(E2+D4)
    Substituting the values from the grid
    18=63(E2+4)18=-6-3(E2+4)
    24=3(E2+4)24=-3(E2+4)
    84=E2-8-4=E2
    E2=12E2=-12
    Answer is Option D
  • Question 4
    1 / -0
    In numbers from 1 to  1001 \ to\  100 the digit "00" appears ____________times.
    Solution
    10,20,30,40,50,60,70,80,90,10010, 20, 30, 40, 50, 60, 70, 80, 90, 100
    Thus the digit 00 appears 1111 times.
  • Question 5
    1 / -0
    The sum of nn terms of the series whose nthn^{th} term is n(n+1)n(n + 1) is equal to.
    Solution
    Tn=n(n+1)=n2+nT_n = n(n+1)=n^2+n
    \therefore Sum == Sn=Tn=n2+n=n(n+1)(2n+1)6+n(n+1)2S_n =\sum T_n = \sum n^2+\sum n=\cfrac{n(n+1)(2n+1)}{6}+\cfrac{n(n+1)}{2}
      =n(n+1)6(2n+1+3)=n(n+1)(n+2)3=\cfrac{n(n+1)}{6}(2n+1+3)=\cfrac{n(n+1)(n+2)}{3}
    Hence, option 'A' is correct.
  • Question 6
    1 / -0
    The sum of nn is equal to
    Solution
    Tn=n(n+1)(n+2)=n3+3n2+2nT_n = n(n+1)(n+2)=n^3+3n^2+2n
    Sn=Tn=n3+3n2+2n\therefore S_n=\sum T_n =\sum n^3+3\sum n^2+2\sum n
                         =(n(n+1)24+3n(n+1)(2n+1)6+2n(n+1)2 =\cfrac{(n(n+1)^2}{4}+3\cfrac{n(n+1)(2n+1)}{6}+2\cfrac{n(n+1)}{2}
                         =14n(n+1)(n2+5n+6)=14n(n+1)(n+2)(n+3) =\cfrac{1}{4}n(n+1)(n^2+5n+6)=\cfrac{1}{4}n(n+1)(n+2)(n+3)
    Hence, option 'B' is correct.
  • Question 7
    1 / -0
    Refer Diagram

    Solution
    The number inside the triangle is equal to the sum of the numbers obtained by multiplying number o either side of the triangle with the base number.
    Thus, 3×1+3×2=93 \times 1 + 3 \times 2 = 9
    5×2+3×2=165 \times 2 + 3 \times 2 = 16
    Hence, the missing term is: 4×1+5×1=94 \times 1 + 5 \times 1 = 9
  • Question 8
    1 / -0
    If CAT is 4848,  Z is 5252 Then what  is TEA equal to ?
    Solution
    Alphanumeric coding type
    Word                               Code value                                 Final code
                                        (alphabet series) 
    CAT                           C=3,A=1,T=20                                24X2=48
                                       3+1+20=24
    Z                                          26                                         24X2=52
    TEA                           T=20,E=5,A=1                                 26X2=52 
                                      20+5+1=26
  • Question 9
    1 / -0
    Choose the correct statement(s):
    AA: Every sequence is a progression.
    BB: Every progression is a sequence.
    Solution
    The difference between a progression and a sequence is that a progression has a specific rule to calculate its next term from its previous term, whereas a sequence can be based on a logical rule like 'a group of prime numbers'.
    Thus, every progression is a sequence but every sequence is not a sequence.
  • Question 10
    1 / -0
    I think of a decimal number. After I have subtracted 2.9 from it, then multiplied by 3 and then added 0.15, I get 10.5. What is the decimal number? 
    Solution
    Let the decimal number be x.
    3(x2.9)+0.15=10.53(x-2.9)+0.15=10.5
    3x8.7+0.15=10.53x-8.7+0.15=10.5
    3x8.7=10.353x-8.7=10.35
    3x=19.053x=19.05
    x=6.35x=6.35.
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