(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...
Question 1 Report
(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 the first three consecutive terms of a Geometric Progression (G.P). The sum of the third and fifth terms of the A.P is 20, find the :
(i) first four terms of the A.P
(ii) sum of the first ten terms of the A.P
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:
\[S_{10}=\frac{10}{2}\big(2a+9d\big)=5\big(5+22.5\big)=5(27.5)=137.5\]
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