Sequences and Series Test - 47

$$\dfrac { 1 }{ { 1+{ 1^{ 2 } }+{ 1^{ 4 } } } } +\dfrac { 2 }{ { 1+{ 2^{ 2 } }+{ 2^{ 4 } } } } +\dfrac { 3 }{ { { 1^{ 3 } }+{ 2^{ 2 } }+{ 3^{ 4 } } } } +.........+\dfrac { { 99 } }{ { 1+{ { 99 }^{ 2 } }+{ { 99 }^{ 4 } } } }  \\ =\sum  _{ n=1 }^{ 99 }{ \dfrac { x }{ { 1+{ x^{ 2 } }+{ x^{ 4 } } } }  } \\ =\sum  _{ n=1 }^{ 99 }{ \dfrac { x }{ { { { \left( { { x^{ 2 } }+1 } \right)  }^{ 2 } }-{ x^{ 2 } } } }  } \\ =\sum  _{ n=1 }^{ 99 }{ \dfrac { { \left( { { x^{ 2 } }+x+1 } \right) -\left( { { x^{ 2 } }-x+1 } \right)  } }{ { 2\left( { { x^{ 2 } }+x+1 } \right) \left( { { x^{ 2 } }-x+1 } \right)  } }  } \\ =\dfrac { 1 }{ 2 } \sum  _{ n=1 }^{ 99 }{ \left( { \dfrac { 1 }{ { { x^{ 2 } }-x+1 } } -\dfrac { 1 }{ { { x^{ 2 } }+x+1 } }  } \right)  } \\ =\dfrac { 1 }{ 2 } \left( { \dfrac { 1 }{ 1 } -\dfrac { 1 }{ 3 } +\dfrac { 1 }{ 3 } -\dfrac { 1 }{ 7 } +\dfrac { 1 }{ 7 } +.........+\dfrac { 1 }{ { { { 99 }^{ 2 } }+99+1 } }  } \right)  \\ =\dfrac { 1 }{ 2 } \left( { 1-\dfrac { 1 }{ { 9901 } }  } \right)  \\ =\dfrac { { 4950 } }{ { 9901 } }  \\ \approx \dfrac { 1 }{ 2 } $$

$$ \\ Hence,\, option\, D\, is\, correct\, answer.$$

$$12,15,21,24,30,33,?,?$$

$$12$$

$$12 + 3 = 15$$

$$15+3\times 2=21$$

$$21+3=24$$

$$24+3\times 2=30$$

$$30+3=33$$

$$33+3\times 2=39$$

$$39+3=42$$

$$\textbf{Step 1: Observe the numbers opposite to each other}$$

                $$ \text{We can see that the number opposite to 315 is 21. Also, } \dfrac{315}{21}=15$$

                $$\text{Hence, we can say the missing number has at least one factor of 15 }$$

                $$\text{We see the missing number is opposite to 15}$$

                $$\text{Also, } 15\times15=225$$

$$\textbf{Step 2: Verification}$$

                $$\text{Let the missing number be x.}$$

                $$\text{ATQ, }$$

                $$\Rightarrow 21+15+x=261$$

                $$\Rightarrow x=261-21-15$$

                $$x=225$$

                $$\text{Hence, it is verified that x=225}$$

         $$6$$             $$18$$            $$28$$            $$35$$

$$\left( 2\times 3 \right) \uparrow $$  $$\left( 2\times 9 \right) \uparrow $$  $$\left( 4\times 7 \right) \uparrow $$  $$\left( 7\times 5 \right) \uparrow $$ $$=110$$

$$\textbf{Step 1: Observe the numbers opposite to each other}$$

                $$ \text{We can see that the number opposite to 196 is 14. Also, } \dfrac{196}{14}=14$$

                $$\text{Hence, we can say the missing number has at least one factor of 14}$$

                $$\text{We see the missing number is opposite to 154}$$

                $$\text{Also, } 14\times11=154$$

                $$\text{Hence, let's assume the missing number is 11}$$

$$\textbf{Step 2: Verification}$$

                $$\text{Let the missing number be x.}$$

                $$\text{According to the question, }$$

                $$\Rightarrow 14+196+x=221$$

                $$\Rightarrow x=221-14-196$$

                $$x=11$$

                $$\text{Hence, it is verified that x=11}$$