Animated Solution for Mathematics - Binomial Theorem: The coefficients of three consecutive terms of $(1+x{)}^{n+5}$ are in the ratio $5:10:14$. Then $n=.........$

The expansion of $(1+x{)}^{n+5}$ involves binomial coefficients that follow the structure of Pascal's Triangle. We are given three consecutive coefficients in the ratio $5:10:14$.

To solve this efficiently, we avoid the cumbersome factorial definition ${}^N{C}_R=\frac{N!}{R!(N-R)!}$. Instead, we utilize the Ratio Property of binomial coefficients.

For any two consecutive binomial coefficients, the ratio is defined as:

This property serves as the master key to unlocking the relationship between the terms without expanding large factorials.

The First Link

Let the three consecutive coefficients be ${}^{n+5}{C}_{r-1}$, ${}^{n+5}{C}_r$, and ${}^{n+5}{C}_{r+1}$. The ratio of the first two is $5:10$, which simplifies to $1:2$.

We set up the ratio as:

Applying the ratio property with $N=n+5$ and $R=r$:

Simplifying this expression leads to our first anchor equation:

The Second Link

Next, we consider the second ratio, $10:14$, which simplifies to $5:7$. We set up the ratio of the second and third terms:

Here, our $R$ is $r+1$. Substituting this into the ratio property:

Cross-multiplying yields $5(n-r+5)=7(r+1)$, which expands to $5n-5r+25=7r+7$. Rearranging the terms gives us our second equation:

We now possess a system of two linear equations:

1) $3r=n+6$

2) $12r=5n+18$

To solve, we multiply the first equation by $4$ to align the $r$ terms:

Equating the two expressions for $12r$:

Solving for $n$, we find the final result: