91Ó°ÊÓ

Find the coefficient of the term containing \(a^{4}\) in the expansion of \((\sqrt{a}-\sqrt{x})^{10}\).

Express each of the sums without using sigma notation. Simplify your answers where possible. $$\sum_{k=4}^{5} k^{2}$$

Use mathematical induction to prove that the formulas hold for all natural numbers \(n\). $$(1+p)^{n} \geq 1+n p, \text { where } p>-1$$

Find the coefficient of the term containing \(a^{4}\) in the expansion of \((3 a-5 x)^{12}\).

Carry out the indicated operations. Express your results in rectangular form for those cases in which the trigonometric functions are readily evaluated without tables or a calculator. $$\left[\sqrt{3}\left(\cos 70_{1}+i \sin 70_{i}\right)\right]^{3}$$

Express each of the sums without using sigma notation. Simplify your answers where possible. $$\sum_{k=2}^{6}(1-2 k)$$

Find the coefficient of the term containing \(y^{8}\) in the expansion of \([(x / 2)-4 y]^{9}\).

Carry out the indicated operations. Express your results in rectangular form for those cases in which the trigonometric functions are readily evaluated without tables or a calculator. $$\left[2^{1 / 5}\left(\cos 63^{\circ}+i \sin 63^{\circ}\right)\right]^{10}$$

Express each of the sums without using sigma notation. Simplify your answers where possible. $$\sum_{n=1}^{3} x^{n}$$

Express each of the sums without using sigma notation. Simplify your answers where possible. $$\sum_{n=1}^{3}(n-1) x^{n-2}$$