Explanation
Step 1: Observing the difference between the adjacent numbers
If we notice the adjacent numbers, we see 4×2−1=7
Similarly, 7×2−1=13
Similarly, 13×2−1=25
Similarly, 25×2−1=49
Step 2: Calculating the missing number
Thus, looking at the above observations, we can conclude the missing number will be:
⇒49×2−1=97
Thus, the missing number is D 97
\textbf{Step 1: Observing the difference between the adjacent numbers}
\text{If we notice the adjacent numbers, we see } 4\times 2 -1 =7
\text{Similarly, } 7 \times 2-1=13
\text{Similarly, } 13\times2 -1=25
\text{Similarly, } 25\times2-1=49
\textbf{Step 2: Calculating the missing number}
\text{Thus, looking at the above observations, we can conclude the missing number will be:}
\Rightarrow 49\times 2-1=97
\textbf{Thus, the missing number is D .}
Given that,
{{C}_{1}}+2.{{C}_{2}}+3.{{C}_{3}}+.............n{{C}_{n}}
=n+2\times \dfrac{n\left( n-1 \right)}{2!}+3\times \dfrac{n\left( n-2 \right)\left( n-3 \right)}{3!}.......n\times 1
=n+\dfrac{n\left( n-1 \right)}{1}+\dfrac{n\left( n-2 \right)\left( n-3 \right)}{2}.......1
=n[1+\dfrac{\left( n-1 \right)}{1}+\dfrac{\left( n-2 \right)\left( n-3 \right)}{2}.......1
Put, n-1=N
=n[1+\dfrac{N}{1}+\dfrac{\left( N+1 \right)\left( N-1 \right)}{2}.......1
=n\left( {}^{N}{{C}_{0}}+{}^{N}{{C}_{1}}+{}^{N}{{C}_{2}}...............{}^{N}{{C}_{N}} \right)
=n{{.2}^{N}}
=n{{.2}^{n-1}}
Hence, this is the answer.
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