Consider the expansion of (1 + x) n. Let p, q, r and s be the coefficients of first, second, nth and (n + 1)th terms respectively. What is (ps + qr) equal to?
Understanding the Binomial Expansion and Coefficients
The question asks us to consider the binomial expansion of $\left(1 + x\right)^n$ and find the value of an expression involving the coefficients of specific terms. The binomial theorem states that the expansion of $\left(1 + x\right)^n$ is given by:
\(\left(1 + x\right)^n = \binom{n}{0} + \binom{n}{1}x + \binom{n}{2}x^2 + \dots + \binom{n}{k}x^k + \dots + \binom{n}{n}x^n\)
The coefficient of the $\left(k+1\right)$-th term in this expansion is given by $\binom{n}{k}$.
Identifying the Coefficients p, q, r, and s
We are given that p, q, r, and s are the coefficients of the first, second, n-th, and $\left(n+1\right)$-th terms respectively. Let's identify the corresponding values of k for each term:
- First term: This is the $\left(1\right)$-th term, so $k+1 = 1$, which means $k=0$. The coefficient is $p = \binom{n}{0}$.
- Second term: This is the $\left(2\right)$-nd term, so $k+1 = 2$, which means $k=1$. The coefficient is $q = \binom{n}{1}$.
- n-th term: This is the $\left(n\right)$-th term, so $k+1 = n$, which means $k=n-1$. The coefficient is $r = \binom{n}{n-1}$.
- $\left(n+1\right)$-th term: This is the $\left(n+1\right)$-th term, so $k+1 = n+1$, which means $k=n$. The coefficient is $s = \binom{n}{n}$.
Calculating the Values of p, q, r, and s
Now let's calculate the specific values of these binomial coefficients:
- $p = \binom{n}{0}$: By definition, $\binom{n}{0} = 1$. So, $p = 1$.
- $q = \binom{n}{1}$: By definition, $\binom{n}{1} = n$. So, $q = n$.
- $r = \binom{n}{n-1}$: Using the property $\binom{n}{k} = \binom{n}{n-k}$, we have $\binom{n}{n-1} = \binom{n}{n - (n-1)} = \binom{n}{1} = n$. So, $r = n$.
- $s = \binom{n}{n}$: By definition, $\binom{n}{n} = 1$. So, $s = 1$.
We can summarize the coefficients in a table:
| Term | Term Number ($\mathbf{k+1}$) | k value | Coefficient | Value |
|---|---|---|---|---|
| First | 1 | 0 | p = $\binom{n}{0}$ | 1 |
| Second | 2 | 1 | q = $\binom{n}{1}$ | n |
| n-th | n | n-1 | r = $\binom{n}{n-1}$ | n |
| (n+1)-th | n+1 | n | s = $\binom{n}{n}$ | 1 |
Evaluating the Expression (ps + qr)
We need to find the value of the expression $\left(ps + qr\right)$. We substitute the values we found for p, q, r, and s:
- $p = 1$
- $q = n$
- $r = n$
- $s = 1$
Substitute these values into the expression:
\(ps + qr = \left(1\right)\left(1\right) + \left(n\right)\left(n\right)\)
Now, perform the multiplication:
\(ps + qr = 1 + n^2\)
Conclusion
The value of $\left(ps + qr\right)$ is $1 + n^2$. We compare this with the given options.
The expression $\left(ps + qr\right)$ is equal to $1 + n^2$.
Revision Table: Binomial Coefficients
| Binomial Coefficient | Formula | Value |
|---|---|---|
| $\binom{n}{0}$ | $\frac{n!}{0!(n-0)!}$ | 1 |
| $\binom{n}{1}$ | $\frac{n!}{1!(n-1)!}$ | n |
| $\binom{n}{k}$ | $\frac{n!}{k!(n-k)!}$ | General formula |
| $\binom{n}{n-1}$ | $\frac{n!}{(n-1)!(n-(n-1))!} = \frac{n!}{(n-1)!1!}$ | n |
| $\binom{n}{n}$ | $\frac{n!}{n!(n-n)!} = \frac{n!}{n!0!}$ | 1 |
Additional Information: Binomial Theorem and Properties
The binomial theorem is a fundamental concept in algebra that provides a formula for expanding powers of binomials. For $\left(1+x\right)^n$, the coefficients are the binomial coefficients $\binom{n}{k}$, which are the numbers in Pascal's triangle.
- The sum of the coefficients in the expansion of $\left(1+x\right)^n$ is obtained by setting $x=1$, which gives $\left(1+1\right)^n = 2^n$. This means $\sum_{k=0}^n \binom{n}{k} = 2^n$.
- The coefficients are symmetric: $\binom{n}{k} = \binom{n}{n-k}$. This property was used to find that $\binom{n}{n-1} = \binom{n}{1} = n$.
- The terms in the expansion are arranged in increasing powers of $x$. The first term has $x^0$, the second term has $x^1$, and the $\left(k+1\right)$-th term has $x^k$.
- The total number of terms in the expansion of $\left(1+x\right)^n$ is $n+1$. This is why the last term is the $\left(n+1\right)$-th term, corresponding to $k=n$.
Understanding these properties helps in quickly identifying the coefficients of specific terms in a binomial expansion.
Important Questions from Binomial Expansion
If $x = \frac{1}{4}$, then the greatest term in the expansion of $(2 + 3x)^{15}$ will be
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