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Class 9 Exploring some more Progressions (Advanced): cheat sheet
CBSE Mathematics. Sums of geometric progressions, the sum of an infinite GP when −1 < r < 1, and the method of differences for sequences whose differences form an AP.
Key ideas
- In a GP each term is r times the one before: t₂ ÷ t₁ = t₃ ÷ t₂ = r.
- When −1 < r < 1, rⁿ shrinks towards 0, so an infinite GP has a finite sum.
- A bouncing ball falls once and then goes up and down: total = H + 2 × (the sum of all the rebound heights).
- The squares of a GP form a GP with ratio r²; so do its odd-numbered and even-numbered terms.
- For a sequence such as 3, 7, 13, 21, write rows of differences until a row is constant.
- Constant second differences mean the nth term is a quadratic in n.
Remember
| Sum of n terms of a GP | Sₙ = a(1 − rⁿ) ÷ (1 − r) = a(rⁿ − 1) ÷ (r − 1), r ≠ 1 |
| Sum to infinity | S = a ÷ (1 − r), only when −1 < r < 1 |
| nth term by differences | tₙ = b + a(n − 1) + d(n − 1)(n − 2)/2 + d₁(n − 1)(n − 2)(n − 3)/6 |
| Sum by differences | Sₙ = bn + a·n(n − 1)/2 + d·n(n − 1)(n − 2)/6 |
| Letters | b = first term; a, d, d₁ = first entries of the 1st, 2nd and 3rd differences |
| 7 + 77 + 777 + … (n terms) | (7/9) × [10(10ⁿ − 1)/9 − n] |
Watch out
- Using a ÷ (1 − r) when r is 1 or more: such a GP has no sum to infinity.
- Counting the first fall of a bouncing ball twice.
- Taking b, a and d from the wrong place: they are the first entries of each row.
Worked example
A square of side 24 cm is cut from card. The midpoints of its sides are joined to make a second square, which is cut and pasted on it, and so on. Find (a) the area of the 8th square (b) the total area of card for the first 8 squares (c) the total area if this went on forever.
Working: the areas form a GP 576, 288, … with r = 1/2; (b) 576(1 − (1/2)⁸) ÷ (1/2); (c) 576 ÷ (1 − 1/2)
Answer: (a) 9/2 cm²; (b) 2295/2 cm²; (c) 1,152 cm²
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