Sequences and series: revision Problem set B One question at a time

The sum of the first n terms of an arithmetic sequence is \(S_n=5n^2-2n\).

1. (a) Write down the values of \(S_1\) and \(S_2\).

   • \(S_1=5(1)^2-2(1)=3\)
   • \(S_2=5(2)^2-2(2)=16\)

2. (b) Write down the values of \(u_1\) and \(u_2\).

   • \(u_1=S_1=3\)
   • \(u_2=S_2-S_1=16-3=13\)

3. (c) Find d, the common difference.

   • \(d=u_2-u_1=13-3=10\)

4. (d) Find \(u_{15}\).

   • \(u_n=3+(n-1)(10)=10n-7\)
   • \(u_{15}=3+14(10)=143\)

5. (e) Find the greatest value of n for which \(u_n<200\).

   • \(10n-7<200\)
   • From the calculator: \(n<20.7\)
   • The greatest whole number is \(n=20\)

6. (f) Find the value of n for which \(S_n=952\).

   • \(5n^2-2n=952\)
   • From the calculator: \(n=14\) or \(n=−13.6\)
   • n is a positive whole number: \(n=14\)

The term \(u_4\) of an arithmetic sequence is 61 and the common difference is \(-7\). Find \(u_1\).

• \(u_4=u_1+3d\Rightarrow 61=u_1+3(-7)\)
• \(u_1=61+21=82\)

An arithmetic sequence has \(u_{4}=5\) and \(u_{12}=61\).

1. (a) Find the common difference.

   • \(u_1+3d=5\) and \(u_1+11d=61\)
   • Subtract: \(8d=56\Rightarrow d=7\)

2. (b) Find the first term.

   • \(u_1=5-3(7)=-16\)

3. (c) Find \(u_{40}\).

   • \(u_{40}=-16+39(7)=257\)

4. (d) Find \(S_{80}\).

   • \(S_{80}=\frac{80}{2}\left(2(-16)+79(7)\right)\)
   • \(S_{80}=20840\)

The term \(u_5\) of an arithmetic sequence is 26 and the common difference is \(-5\). The first and third terms of this sequence are the first two terms of a geometric sequence. Find the sixth term of the geometric sequence.

• \(u_1=26-4(-5)=46\), \(u_3=46+2(-5)=36\)
• \(r=\frac{36}{46}=\frac{18}{23}\)
• \(u_6=46\left(\frac{18}{23}\right)^5=13.5\)

The term \(u_3\) of an arithmetic sequence is 21 and the common difference is \(-5\). The first and fourth terms of this sequence are the first two terms of an infinite geometric series. Find the sum to infinity of the geometric series.

• \(u_1=21-2(-5)=31\), \(u_4=31+3(-5)=16\)
• \(r=\frac{16}{31}\), and \(|r|<1\)
• \(S_\infty=\frac{31}{1-\frac{16}{31}}=\frac{961}{15}=64.1\)

A student saves 15 euros in the first week. Each week after that the student saves 4 euros more than the week before.

1. (a) Find the amount saved in week 30.

   • \(u_1=15,\ d=4\)
   • \(u_{30}=15+29(4)=131\)

2. (b) Find the total saved in the first 30 weeks.

   • \(S_{30}=\frac{30}{2}\left(2(15)+29(4)\right)=2190\)

3. (c) The student wants to have saved at least 3000 euros. Find the least number of weeks needed.

   • \(S_n=\frac n2\left(2(15)+(n-1)(4)\right)\ge3000\)
   • From the calculator: \(S_n=3000\) at \(n=35.6\)
   • The least whole number is \(n=36\)

A runner runs 12 km in the first week of a training plan. Each week the runner runs 8% more than the week before. Give your answers to 3 significant figures.

1. (a) Find the distance run in week 10.

   • Geometric: \(u_1=12,\ r=1.08\)
   • \(u_{10}=12(1.08)^{9}=24\)

2. (b) Find the total distance run in the first 10 weeks.

   • \(S_{10}=\frac{12\left(1.08^{10}-1\right)}{1.08-1}=174\)

A forest team plants 72 trees in the first year. Each later year it plants 75% as many as the year before, and this continues for ever. Find the total number of trees planted.

• \(u_1=72,\ r=0.75\), and \(|r|<1\)
• \(S_\infty=\frac{72}{1-0.75}=288\)

An arithmetic sequence has \(S_{8}=104\) and \(S_{16}=400\). Find the first term and the common difference.

• \(\frac{8}{2}(2u_1+7d)=104\), so \(2u_1+7d=26\)
• \(\frac{16}{2}(2u_1+15d)=400\), so \(2u_1+15d=50\)
• Subtract: \(8d=24\), so \(d=3\)
• \(2u_1=26-7(3)\), so \(u_1=2.5\)