N2020 P1 Q8

N2020 P1 Q8

Junior College 2
12 marks
Free
  1. The 1st term of an arithmetic series is \(4\) and the 5th term is \(10\).
    1. Find the 30th term of this series.[2]
    2. Find the sum of the 21st term to the 50th term inclusive of this series.[3]
  2. The 1st term of a geometric series is \(4\) and the 5th term is \(1.6384\), where the common ratio is positive.
    1. Find the sum to infinity of this series.[2]
    2. Given that the sum of the first \(n\) terms is greater than \(19.6\), show that \(0.8^n<0.02\). Hence find the smallest possible value of \(n\).[5]

Default solution

(a) \(d=(10-4)/4=1.5\). Hence \(u_{30}=4+29(1.5)=47.5\).

\(u_{21}=34\) and \(u_{50}=77.5\), so the sum is \(\frac{30}{2}(34+77.5)=1672.5\).

(b) \(4r^4=1.6384\) and \(r>0\), giving \(r=0.8\). The sum to infinity is \(4/(1-0.8)=20\).

\(S_n=20(1-0.8^n)>19.6\) exactly when \(0.8^n<0.02\). Taking logarithms gives \(n>\frac{\ln0.02}{\ln0.8}\approx17.53\), so the least integer is \(18\).

Answer:(a)(i) \(47.5\), (ii) \(1672.5\); (b)(i) \(20\), (ii) \(18\)

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