Elaborate Solution for Binomial Coefficient Ratios

1. Key Concept: The General Term in Binomial Expansion

The binomial theorem states that the expansion of $(a+b)^n$ is given by: $(a+b)^n = \sum_{k=0}^{n} {^nC_k} a^{n-k} b^k$ The general term, often denoted as $T_{k+1}$ (the $(k+1)$-th term from the beginning), is: $T_{k+1} = {^nC_k} a^{n-k} b^k$ In this problem, we are dealing with the expansion of $(1+2x)^n$. Here, $a=1$ and $b=2x$. Substituting these into the general term formula: $T_{k+1} = {^nC_k} (1)^{n-k} (2x)^k = {^nC_k} 2^k x^k$ The coefficient of the $(k+1)$-th term, $T_{k+1}$, is therefore ${^nC_k} 2^k$.

2. Defining the Coefficients of Three Consecutive Terms

Let the three consecutive terms be $T_r$, $T_{r+1}$, and $T_{r+2}$.

• The term $T_r$ corresponds to $k = r-1$ in our general term formula. So, its coefficient is $C_{r-1} = {^nC_{r-1}} 2^{r-1}$.
• The term $T_{r+1}$ corresponds to $k = r$ in our general term formula. So, its coefficient is $C_r = {^nC_r} 2^r$. This will be the middle term.
• The term $T_{r+2}$ corresponds to $k = r+1$ in our general term formula. So, its coefficient is $C_{r+1} = {^nC_{r+1}} 2^{r+1}$.

We are given that the ratio of these three consecutive coefficients is $2 : 5 : 8$. So, we can write: ${^nC_{r-1}} 2^{r-1} : {^nC_r} 2^r : {^nC_{r+1}} 2^{r+1} = 2 : 5 : 8$

3. Forming Equations from the Ratios

We can form two separate equations from this given ratio.

Equation 1: Ratio of the first two coefficients $\frac{{^nC_{r-1}} 2^{r-1}}{{^nC_r} 2^r} = \frac{2}{5}$ To simplify the ratio of binomial coefficients, we use the identity: $\frac{{^nC_k}}{{^nC_{k-1}}} = \frac{n-k+1}{k}$. Therefore, $\frac{{^nC_{r-1}}}{{^nC_r}} = \frac{r}{n-r+1}$. Substitute this back into the equation: $\frac{r}{n-r+1} \cdot \frac{2^{r-1}}{2^r} = \frac{2}{5}$ $\frac{r}{n-r+1} \cdot \frac{1}{2} = \frac{2}{5}$ Multiply both sides by 2: $\frac{r}{n-r+1} = \frac{4}{5}$ Cross-multiply to get rid of the denominators: $5r = 4(n-r+1)$ $5r = 4n - 4r + 4$ Rearrange the terms to get our first linear equation: $9r = 4n + 4 \quad \text{(1)}$

• Tip: A common mistake is to get the indices for the combination ratio inverted or to make an error in simplifying the powers of 2. Always simplify the constant terms (like $2^r$) separately from the combinations.

Equation 2: Ratio of the second and third coefficients $\frac{{^nC_r} 2^r}{{^nC_{r+1}} 2^{r+1}} = \frac{5}{8}$ Using a similar identity for binomial coefficients: $\frac{{^nC_k}}{{^nC_{k+1}}} = \frac{k+1}{n-k}$. So, $\frac{{^nC_r}}{{^nC_{r+1}}} = \frac{r+1}{n-r}$. Substitute this back into the equation: $\frac{r+1}{n-r} \cdot \frac{2^r}{2^{r+1}} = \frac{5}{8}$ $\frac{r+1}{n-r} \cdot \frac{1}{2} = \frac{5}{8}$ Multiply both sides by 2: $\frac{r+1}{n-r} = \frac{10}{8} = \frac{5}{4}$ Cross-multiply: $4(r+1) = 5(n-r)$ $4r + 4 = 5n - 5r$ Rearrange the terms to get our second linear equation: $9r = 5n - 4 \quad \text{(2)}$

4. Solving the System of Equations

Now we have a system of two linear equations with two variables ($n$ and $r$):

1. $9r = 4n + 4$
2. $9r = 5n - 4$ Since both equations are equal to $9r$, we can set them equal to each other: $4n + 4 = 5n - 4$ Now, solve for $n$: $4 + 4 = 5n - 4n$ $8 = n$ So, $n=8$.

Now, substitute the value of $n=8$ into either Equation (1) or (2) to find $r$. Let's use Equation (1): $9r = 4(8) + 4$ $9r = 32 + 4$ $9r = 36$ $r = \frac{36}{9}$ $r = 4$ Thus, we have found $n=8$ and $r=4$.

5. Identifying and Calculating the Middle Term's Coefficient

The problem asks for the coefficient of the term which is in the middle of those three consecutive terms. As established in Section 2, the three consecutive terms are $T_r$, $T_{r+1}$, and $T_{r+2}$. The middle term among these is $T_{r+1}$. The coefficient of $T_{r+1}$ is ${^nC_r} 2^r$. Substitute the values $n=8$ and $r=4$: $\text{Coefficient} = {^8C_4} 2^4$ First, calculate the binomial coefficient ${^8C_4}$: ${^8C_4} = \frac{8!}{4!(8-4)!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1}$ ${^8C_4} = \frac{8 \times 7 \times 6 \times 5}{24} = 70$ Next, calculate $2^4$: $2^4 = 2 \times 2 \times 2 \times 2 = 16$ Finally, multiply these values to get the coefficient: $\text{Coefficient} = 70 \times 16$ $\text{Coefficient} = 1120$

6. Summary and Key Takeaway

By correctly identifying the general term of the binomial expansion and setting up a system of equations based on the given ratio of consecutive coefficients, we were able to determine the values of $n$ and $r$. This allowed us to find the specific middle term's coefficient as requested. The key was careful application of the binomial coefficient identities and systematic algebraic solving.