(a) The first term of an Arithmetic Progression(AP) is 3 and the common difference is 4. Find the sum of the first 28 terms. (b) Given that \(x = \frac{2m}{...
Question 1 Report
(a) The first term of an Arithmetic Progression(AP) is 3 and the common difference is 4. Find the sum of the first 28 terms.
(b) Given that \(x = \frac{2m}{1 - m^{2}}\) and \(y = \frac{2m}{1 + m}\), express 2x - y in terms of m in the simplest form.
(c) The angles of pentagon are x°, 2x°, 3x°, 2x° and (3x - 10)°. Find the value of x.
Answer Details
(a) AP with \(a=3,\;d=4,\;n=28\). \[S_{28}=\frac{n}{2}\bigl(2a+(n-1)d\bigr)=14\bigl(6+27\times4\bigr)=14\times114=1596.\]
(b) \(x=\dfrac{2m}{1-m^{2}}=\dfrac{2m}{(1-m)(1+m)}\) and \(y=\dfrac{2m}{1+m}\). \[2x-y=\frac{4m}{(1-m)(1+m)}-\frac{2m}{1+m}=\frac{4m-2m(1-m)}{(1-m)(1+m)}=\frac{2m+2m^{2}}{(1-m)(1+m)}=\frac{2m(1+m)}{(1-m)(1+m)}=\frac{2m}{1-m}.\]
(c) Interior angles of a pentagon sum to \(540^{\circ}\): \[x+2x+3x+2x+(3x-10)=540\Rightarrow 11x-10=540\Rightarrow 11x=550\Rightarrow x=50.\]
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