(a) If \(f(x) = \int (4x - x^{2}) \mathrm {d} x\) and f(3) = 21, find f(x). (b) The second, fourth and eigth terms of an Arithmetic Progression (A.P) form t...

Answer Details

(a) \(f(x)=\displaystyle\int(4x-x^{2})\,dx=2x^{2}-\dfrac{x^{3}}{3}+C\).

Use \(f(3)=21\): \(2(9)-\dfrac{27}{3}+C=18-9+C=9+C=21\Rightarrow C=12\).

\[f(x)=2x^{2}-\frac{x^{3}}{3}+12\]

(b) Let the AP have first term \(a\) and common difference \(d\). The 2nd, 4th, 8th terms are \(a+d,\ a+3d,\ a+7d\) and form a GP:

\[(a+3d)^{2}=(a+d)(a+7d)\]

\[a^{2}+6ad+9d^{2}=a^{2}+8ad+7d^{2}\Rightarrow 2d^{2}-2ad=0\Rightarrow 2d(d-a)=0\]

Since \(d\neq0\), \(a=d\).

Sum of 3rd and 5th terms is \(20\): \((a+2d)+(a+4d)=2a+6d=20\Rightarrow a+3d=10\).

With \(a=d\): \(4d=10\Rightarrow d=2.5,\ a=2.5\).

(i) First four terms: \(2.5,\ 5,\ 7.5,\ 10\).

(ii) Sum of first ten terms:

\[S_{10}=\frac{10}{2}\big(2a+9d\big)=5\big(5+22.5\big)=5(27.5)=137.5\]