Self Studies

Sequences and S...

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  • Question 1
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    The value of r=1630(r+2)(r3)  \displaystyle \sum_{r = 16}^{30}{ (r + 2)(r - 3) }  is equal to :

  • Question 2
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    The sum of the following series 1+6+9(12+22+32)7+12(12+22+32+42)9+15(12+22+....+52)11+.....1+6+\dfrac{9(1^2+2^2+3^2)}{7}+\dfrac{12(1^2+2^2+3^2+4^2)}{9}+\dfrac{15(1^2+2^2+....+5^2)}{11}+..... up to 1515 terms is:

  • Question 3
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    Let AA be the sum of the first 2020 terms and BB be the sum of the first 4040 terms of the series 12+2.22+32+2.42+52+2.62+.....1^{2} + 2.2^{2} + 3^{2} + 2.4^{2} + 5^{2} + 2.6^{2} + ..... If B2A=100λB - 2A = 100\lambda, then λ\lambda is equal to

  • Question 4
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    If the sum of the first 15 terms of the series 343+1123+2143+33+3343+...........\lgroup \frac{3}{4} \rgroup^3 + \lgroup 1\frac{1}{2} \rgroup^3 + \lgroup 2\frac{1}{4} \rgroup^3 + 3^3 + \lgroup 3\frac{3}{4} \rgroup^3 + ........... is equal to 225 k. then k is equal to:

  • Question 5
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    Let  Sk=1+2+3++kk.S _ { k } = \dfrac { 1 + 2 + 3 + \ldots + k } { k } .  If   S12+S22++S102=512A{ S }_{ { 1 } }^{ { 2 } }+{ S }_{ { 2 } }^{ { 2 } }++{ S }_{ { 10 } }^{ { 2 } }=\dfrac { 5 }{ 12 } { A }  then  AA  is equal to :

  • Question 6
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    The sum V1+V2++Vn\mathrm{V}_{1}+\mathrm{V}_{2}+\ldots+\mathrm{V}_{\mathrm{n}} is 

  • Question 7
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    Let a1, a2, a3, , a11\mathrm{a}_{1},\ \mathrm{a}_{2},\ \mathrm{a}_{3},\ \ldots,\ \mathrm{a}_{11} be real numbers satisfying a1=15,272a2>0\mathrm{a}_{1}=15,27-2\mathrm{a}_{2}>0 and ak=2ak1ak2\mathrm{a}_{\mathrm{k}}=2\mathrm{a}_{\mathrm{k}-1}-\mathrm{a}_{\mathrm{k}-2} 

    for k=3,4, ,11.\mathrm{k}=3,4,\ \ldots, 11. If a12+a22++a11211=90\displaystyle \frac{a_{1}^{2}+a_{2}^{2}+\ldots+a_{11}^{2}}{11}=90, then the value of a1+a2++a1111\displaystyle \frac{a_{1}+a_{2}+\ldots+a_{11}}{11} is equal to

  • Question 8
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    x(x+1x)2+(x2+1x2)2+(x3+1x3)2x\left (x + \dfrac {1}{x}\right )^{2} + \left (x^{2} + \dfrac {1}{x^{2}}\right )^{2} + \left (x^{3} + \dfrac {1}{x^{3}}\right )^{2} .... upto nn terms is

  • Question 9
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    Which letter will be the fifth from the right if the first and second, the third and fourth and so on are interchanged among each other in the word "COMPANIONATE"?

  • Question 10
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    Mean of the first nn terms of the A.P. a,(a+d),(a+2d),........a, (a + d), (a + 2d), ........ is

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