Key Concepts Used

This problem leverages fundamental identities related to sums of binomial coefficients, particularly:

1. The sum of all binomial coefficients: $\sum_{r=0}^{n} {}^{n}C_{r} = {}^{n}C_0 + {}^{n}C_1 + \dots + {}^{n}C_n = 2^n$
2. The sum involving the product of $r$ and a binomial coefficient: $\sum_{r=0}^{n} r \cdot {}^{n}C_{r} = n \cdot 2^{n-1}$ This identity can be derived by noting that $r \cdot {}^{n}C_r = r \cdot \frac{n!}{r!(n-r)!} = \frac{n!}{(r-1)!(n-r)!} = n \cdot \frac{(n-1)!}{(r-1)!((n-1)-(r-1))!} = n \cdot {}^{n-1}C_{r-1}$. Therefore, $\sum_{r=0}^{n} r \cdot {}^{n}C_r = \sum_{r=1}^{n} n \cdot {}^{n-1}C_{r-1} = n \sum_{k=0}^{n-1} {}^{n-1}C_k = n \cdot 2^{n-1}$.
3. The Greatest Integer Function (Floor Function) denoted by $[x]$, which gives the largest integer less than or equal to $x$.

Step-by-step Derivation of the Sum

The given series is ${}^{n}C_0 + 3 \cdot {}^{n}C_1 + 5 \cdot {}^{n}C_2 + 7 \cdot {}^{n}C_3 + \dots$ This sum consists of $(n+1)$ terms. Let's identify the pattern for the general term. The $r$-th term (starting with $r=0$) has a binomial coefficient ${}^{n}C_r$ and a coefficient that forms an arithmetic progression: $1, 3, 5, \dots$. The general term of an arithmetic progression is $a + (r)d$. Here, the first term $a=1$ and the common difference $d=2$. So, the coefficient for ${}^{n}C_r$ is $1 + r \cdot 2 = 2r+1$.

Thus, the general term of the series, $T_r$, can be written as $T_r = (2r+1) {}^{n}C_r$. The sum $S$ is then given by: $S = \sum_{r=0}^{n} (2r+1) {}^{n}C_r$

To simplify this sum, we can split the general term into two parts: $S = \sum_{r=0}^{n} (2r \cdot {}^{n}C_r + 1 \cdot {}^{n}C_r)$ This can be further separated into two distinct sums: $S = \sum_{r=0}^{n} 2r \cdot {}^{n}C_r + \sum_{r=0}^{n} {}^{n}C_r$

Now, we evaluate each sum using the key identities:

1. Evaluate the first sum: $\sum_{r=0}^{n} 2r \cdot {}^{n}C_r$ $2 \sum_{r=0}^{n} r \cdot {}^{n}C_r$ Using the identity $\sum_{r=0}^{n} r \cdot {}^{n}C_r = n \cdot 2^{n-1}$, we get: $2 \cdot (n \cdot 2^{n-1})$ $= n \cdot 2^1 \cdot 2^{n-1} = n \cdot 2^{n-1+1} = n \cdot 2^n$

2. Evaluate the second sum: $\sum_{r=0}^{n} {}^{n}C_r$ Using the identity $\sum_{r=0}^{n} {}^{n}C_r = 2^n$, we get: $2^n$

3. Combine the results: Substituting these back into the expression for $S$: $S = n \cdot 2^n + 2^n$ We can factor out $2^n$: $S = 2^n (n+1)$

Solving for n

We are given that the sum $S$ is equal to $2^{100} \cdot 101$. So, we can set up the equation: $2^n (n+1) = 2^{100} \cdot 101$ By comparing the terms on both sides of the equation, we can directly deduce the value of $n$. The base of the power is 2 on both sides, and the term $(n+1)$ corresponds to $101$. Therefore, we have: $n = 100$

Evaluating the Final Expression

The problem asks for the value of $2\left[ \frac{n-1}{2} \right]$. Substitute the value $n=100$ that we just found: $2\left[ \frac{100-1}{2} \right]$ $= 2\left[ \frac{99}{2} \right]$ $= 2\left[ 49.5 \right]$ Now, apply the greatest integer function. The greatest integer less than or equal to $49.5$ is $49$. $= 2 \cdot 49$ $= 98$

Tips for Success

• Recognize Patterns: Always look for patterns in the coefficients of binomial series. Here, $(2r+1)$ was key.
• Split Complex Sums: If a sum has a complex general term, try to split it into simpler parts that correspond to known identities.
• Master Binomial Identities: Familiarize yourself with common binomial coefficient identities, especially those involving sums like $\sum {}^{n}C_r$ and $\sum r \cdot {}^{n}C_r$.
• Understanding Greatest Integer Function: Remember that $[x]$ always rounds down to the nearest integer. Pay careful attention to negative numbers as well, though not relevant here.

Summary and Key Takeaway

This problem beautifully illustrates how to tackle sums of binomial coefficients when they are multiplied by terms forming an arithmetic progression. The strategy involves expressing the series in summation notation, breaking the general term into components that align with standard binomial sum identities, and then solving for the unknown variable. Finally, careful application of the greatest integer function is required to reach the correct numerical answer.