Show that the power of $x$ with the largest coefficient in the polynomial $\left( 1 + \frac { 2 x } { 3 } \right) ^ { 20 }$ is 8 , i.e., if we write the given polynomial as $\sum _ { i } a _ { i } x ^ { i }$ then the largest coefficient $a _ { i }$ is $a _ { 8 }$.
(i) By the binomial theorem $(\sqrt{2} + 1)^{10} = \sum_{i=0}^{10} C_i (\sqrt{2})^i$, where $C_i$ are appropriate constants. Write the value of $i$ for which $C_i (\sqrt{2})^i$ is the largest among the 11 terms in this sum. (ii) For every natural number $n$, let $(\sqrt{2} + 1)^n = p_n + \sqrt{2} q_n$, where $p_n$ and $q_n$ are integers. Calculate $\lim_{n \rightarrow \infty} \left(\frac{p_n}{q_n}\right)^{10}$.
Notice that the quadratic polynomial $p(x) = 1 + x + \frac{1}{2}x(x-1)$ satisfies $p(j) = 2^{j}$ for $j = 0, 1$ and $2$. A polynomial $q(x)$ of degree 7 satisfies $q(j) = 2^{j}$ for $j = 0, 1, 2, 3, 4, 5, 6, 7$. Find the value of $q(10)$.
In the expansion of the polynomial $2 ( x + a ) ^ { n }$, the coefficient of $x ^ { n - 1 }$ and the coefficient of $x ^ { n - 1 }$ in the expansion of the polynomial $( x - 1 ) ( x + a ) ^ { n }$ are equal. Find the maximum value of $a n$ for all ordered pairs $( a , n )$ satisfying this condition. (Here, $a$ is a natural number and $n$ is a natural number with $n \geqq 2$.) [4 points]
In the expansion of the polynomial $( x - a ) ^ { 5 }$, when the sum of the coefficient of $x$ and the constant term is 0, what is the value of the positive constant $a$? [3 points] (1) 1 (2) 2 (3) 3 (4) 4 (5) 5
In the expansion of the polynomial $( x + a ) ^ { 7 }$, when the coefficient of $x ^ { 4 }$ is 280, what is the coefficient of $x ^ { 5 }$? (where $a$ is a constant) [3 points] (1) 84 (2) 91 (3) 98 (4) 105 (5) 112
In the expansion of the polynomial $( x + a ) ^ { 6 }$, if the coefficient of $x ^ { 4 }$ is 60, what is the value of the positive number $a$? [3 points] (1) 1 (2) 2 (3) 3 (4) 4 (5) 5
What is the coefficient of $x ^ { 4 }$ in the expansion of $\left( x + \frac { 2 } { x } \right) ^ { 8 }$? [3 points] (1) 108 (2) 112 (3) 116 (4) 120 (5) 124
In the expansion of $\left( 2 x + \frac { 1 } { x ^ { 2 } } \right) ^ { 4 }$, what is the coefficient of $x$? [3 points] (1) 16 (2) 20 (3) 24 (4) 28 (5) 32
9. In the expansion of the binomial $( x + 1 ) ^ { 10 }$, if a term is chosen at random, the probability that the coefficient of that term is odd is $\_\_\_\_$. (Express the result as a fraction)
4. In the expansion of the binomial $( x + 1 ) ^ { n } \left( n \in N _ { + } \right)$, the coefficient of $x ^ { 2 }$ is 15. Then $n =$ A. 4 B. 5 C. 6 D. 7
12. The coefficient of $x ^ { 8 }$ in the expansion of $\left( x ^ { 3 } + \frac { 1 } { 2 \sqrt { x } } \right) ^ { 5 }$ is $\_\_\_\_$ (answer with numerals).