Geometric progressions, finite sums, the condition for convergence, the sum to infinity, recurring decimals and real-world applications.
5 narrated scenes. Press play, listen and read along — the points appear as they are explained. Use ‹ Prev / Next › or the scene bar to jump to any scene.
A ball loses a fraction of its height at every bounce. Add up infinitely many bounces and you get a finite distance — that is a convergent geometric series.
Narration uses your device’s built-in voice. No sound? Switch to “Captions only” — the captions advance on their own.
Change the common ratio and see when the series converges.
Copy these into your notebook in your own words — that is how they stick.
Tₙ = arⁿ⁻¹; r = Tₙ₊₁/Tₙ.
Sₙ = a(rⁿ − 1)/(r − 1) = a(1 − rⁿ)/(1 − r).
S∞ = a/(1 − r), only if |r| < 1.
rⁿ → 0 iff −1 < r < 1; otherwise diverges.
Divide: r^{m−n} = Tₘ/Tₙ.
Use logarithms: rⁿ > k → n lg r > lg k (watch the sign if r < 1).
Write as a GP and use S∞.
AP: always divergent. GP: convergent iff |r| < 1.
3 quick games on this lesson. Earn XP and badges as you play — your best score for each game is saved. Games are for practice and do not change your exam readiness. More games in the Additional Mathematics Arcade.
8 items modelled on real CSEC Paper 1 questions. Choose one answer for each, then check. Aim for 80% or better. Work each item on paper first.
Write full answers, then reveal the mark scheme and tick the points you earned (27 marks in total). Show all your working — method marks count. Your answers are saved on this device.
Question 1 [15 marks]
The first three terms of a geometric progression are 3, 6 and 12.
(a) Find the 10th term and the sum of the first 10 terms. [4]
(b) Find the least number of terms for which the sum exceeds 10 000. [4]
(c) A second GP has first term 8 and common ratio 1/2. Show that its sum to infinity is 16. [3]
(d) Express the recurring decimal 0.272727… as a fraction in its lowest terms. [4]
Question 2 [12 marks]
A ball is dropped from a height of 10 m. After each bounce it rises to 60% of the height from which it fell.
(a) Show that the heights reached after the first and second bounces are 6 m and 3.6 m. [2]
(b) Write the height after the nth bounce and find it for n = 8, to 3 decimal places. [3]
(c) Find the total distance the ball travels from the moment it is dropped until it comes to rest. [5]
(d) Explain why the total distance is finite. [2]
Find these in your own past-paper booklets for extra timed practice. Items above are written in the style of these papers; any adapted from a specific question say so.