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# Series and Binomial Theorem

See the attached.

Q2 a) If the x3, the 3rd term of a series is 3, and x17, the 17th term is -27, state the series.

Find if the series has a common difference.

b) Coolant hose and nozzle design that to be used on CNC machines, has to ensure that adequate coolant is delivered to the cutter/workpiece interface to maintain surface finish and minimise tool wear. The designer of the nozzle is required to design a series of apertures with a minimum diameter of 5 mm and maximum diameter of 20 mm and with 4 intermediate sizes such that they form a non-linear progression. What diameters would you recommend?

Q3 a) Use the Binomial Theorem to expand (3 - x)5.

b) Check the answer you found for question 3a) by using Pascal's Triangle method.

Q4 Use Maclaurin's Series to find the first four non-zero terms of (cosh &#61638;)

Q5 Using standard power series definitions, state a series for ex. sin x (first three non-zero terms).

Q6 a) Solve 3 sinh &#61537; + 2 cosh &#61537; = 7

b) In a capacitive circuit, the instantaneous voltage across the capacitor is given by:

v = V(1 - e-t/CR)

where V is the initial supply voltage, R the resistance, C the capacitance and t the time in seconds from the instant of connecting the supply voltage. If V = 200 volts, R = 10000 ohms and C = 20 x 10-6 farads, how long will it be before the instantaneous voltage is 100 volts?

Q7 Find the value of (6a3 + 3)/(2a3 - 5) as n &#61614; &#61605;

#### Solution Summary

This is a set of questions regarding a series, minimization, the binomial theorem, Pascal's triangle, Maclaurin's series, power series, and a capacitative circuit.

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