Exercises 11 (Arithmetic and geometric progressions)

  1. For the two progressions below, determine the \(20^{\mathrm{th}}\) term:

    1. \(3,\,7.5,\,18.75,\dots\)
    2. \(6,\,18,\,30,\dots\)

  2. Determine the common difference for an AP where the sum for the first 10 terms of the progression is \(233.75\), given that the first term is \(6.5\).

  3. How many terms of the progression \(4,10,16,\dots\) must be taken so that the sum is equal to \(602\)?

  4. A machine is required to have 5 speeds, the lowest being 60 rev/min and the highest 680 rev/min. State the complete range of speeds i) in AP and ii) in GP.

  5. A drilling machine has 10 speeds arranged in GP and operates at a surface cutting speed of \(15\,\mathrm{m/s}\). The smallest drill bit has diameter \(6\,\mathrm{mm}\) and the largest drill bit has diameter \(18\,\mathrm{mm}\). Determine the complete range of speeds. (Speed in \(\mathrm{rev}\cdot\mathrm{s}^{-1}\) = surface cutting speed in \(\mathrm{m}\cdot\mathrm{s}^{-1}/\)drill bit circumference in \(\mathrm{m}\).)

  6. A tie rod \(5\,\mathrm{m}\) long is made such that the cross-sectional areas at equal distances are a geometric progression. The area at the smaller end is \(10\,\mathrm{mm}^2\) whilst the area one tenth of the way down the rod is \(25\,\mathrm{mm}^2\). Calculate the area of the cross section at the larger end.

  7. A body falling freely falls \(4.9\,\mathrm{m}\) in the \(1^{\mathrm{st}}\) second, \(14.7\,\mathrm{m}\) in the \(2^{\mathrm{nd}}\) second, \(24.5\,\mathrm{m}\) in the \(3^{\mathrm{nd}}\) second and so on. Determine:

    1. How far it falls in the \(10^{\mathrm{th}}\) second.
    2. The total distance fallen in \(10\) seconds.

  8. If £50 is saved in a certain year and each year thereafter £5 more is saved than in the previous year, after how many years will the total equal £1950, excluding any interest?

Solutions: 1. (i) \(109\times 10^6\); (ii) \(234\); 2. \(d=3.75\); 3. \(14\); 4. (i) \(60,215,370,525,680\); (ii) \(60,110,202,371,680\); 5. \(265,299,338,382,432,488,552,623,704,796\); 6. \(95\times 10^3\,\mathrm{mm}^2\); 7. (a) \(93.1\,\mathrm{m}\); (b) \(490\,\mathrm{m}\); 8. \(20\) years