Exploring some more Progressions (Advanced)

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	S_n = 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	t_n = b + a(n − 1) + d(n − 1)(n − 2)/2 + d₁(n − 1)(n − 2)(n − 3)/6
Sum by differences	S_n = 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